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Metastable dynamics of convection-diffusion-reaction equations Sun, Xiaodi 1998

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M E T A S T A B L E D Y N A M I C S OF C O N V E C T I O N - D I F F U S I O N - R E A C T I O N E Q U A T I O N S B y Xiaodi Sun B. Sc. (Computational Mathematics) Nanjing University, P.R. China, 1984 M . Sc. (Numerical Analysis) Nanjing University, P.R. China, 1987 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F M A T H E M A T I C S I N S T I T U T E O F A P P L I E D M A T H E M A T I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 1998 © Xiaodi Sun, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mfvthe VA(A\ I ^ The University of British Columbia Vancouver, Canada Date (7(4- 14 ,. fW DE-6 (2/88) A b s t r a c t Metastable dynamics, which qualitatively refers to physical processes that involve an extremely slow approach to their final equilibrium states, is often associated with singu-larly perturbed convection-diffusion-reaction equations. A problem exhibits metastable behavior when the approach to equilibrium occurs on a time-scale of order 0(e?) , where C > 0 and e is the singular perturbation parameter. The studies of these mathematical models are not only significant in their own right, but also useful in simulating and ex-plaining observed physical phenomena and exploring possibly certain unknown ones. A typical common characteristic associated with these convection-diffusion-reaction equa-tions is that the linearized operator is exponentially ill-conditioned. By exponential ill-conditioning we mean that the linearized operator has an exponentially small eigen-value. As a result, conventional analytical methods and numerical schemes may fail to provide accurate information about the metastable behavior. This thesis is concerned with developing a systematic and robust approach based on asymptotic and numerical methods to quantify the dynamic metastability associated with various problems. Using the asymptotic method called the projection method which was originated by Ward in [108], we have succeeded in deriving ordinary differential equations (ODEs) or differential algebraic equations (DAEs) which characterize the metastable patterns for several problems, including, the phase separation of a binary alloy modeled by the viscous Cahn-Hilliard equation, the upward propagation of a flame front in a vertical channel modeled by the Mikishev-Rakib-Sivashinsky equation, and two problems in slowly varying geometries. The main role of our numerical method called the transverse method of lines is to give a numerical justification of these ODEs/DAEs and to provide ii useful information about the metastable solutions in their transient phases and collapse phases during which our asymptotic method fails. From the numerical point of view, little is known of the nature concerning the con-vergence and stability of any numerical scheme that computes metastable behavior, as a result of the exponential ill-conditioning of the linearized operator. In this thesis, several finite difference schemes and their convergence are analyzed rigorously for a boundary layer resonance problem. Our results from this problem are shown numerically to be also valid for other nonlinear metastable problems and some guidelines in designing effective numerical schemes are provided. The analytical and numerical results show that our approach is a powerful and general tool to quantitatively study the metastable patterns in various physical problems. In addition, the metastable behavior revealed by our analysis appears to be also rather interesting from the viewpoint of physical applications. in Table of Contents Abstract i i List of Tables v i i List of Figures x Acknowledgements xi i 1 Introduction 1 1.1 Burgers Equation for Viscous Shocks 4 1.2 The Projection Method 8 1.3 The Numerical Method T M O L 13 1.4 Contribution of This Thesis 15 2 Metastability in Upward Propagating Flame 19 2.1 A Generalized Burgers Equation 19 2.2 The Equilibrium Problem 28 2.2.1 A n Application of the Projection Method 31 2.3 Asymptotics and Numerics for the Principal Eigenpair 33 2.3.1 Asymptotics for Principal Eigenfunction 33 2.3.2 Asymptotics for the Principal Eigenvalue 35 2.3.3 Numerics for the Principal Eigenvalue 38 2.4 Derivation of the Metastable Dynamics 41 2.5 Comparison of Asymptotic and Numerical Results 45 iv 3 Metastability in Slowly Varying Geometry Problems 55 3.1 Convection-Diffusion-Reaction Equations in T h i n Domains 55 3.2 A Generalized Ginzburg-Landau Equation 59 3.2.1 The Eigenvalue Analysis 60 3.2.2 The Metastabilty Analysis 62 3.2.3 Comparison of Asymptotic and Numerical Results 65 3.3 A Burgers-like Convection-Diffusion-Reaction Equation 72 3.3.1 The Eigenvalue Analysis 74 3.3.2 The Metastability Analysis 76 3.3.3 Comparison of Asymptotic and Numerical Results 80 4 Phase Separation Models in One Spatial Dimension 90 4.1 The Viscous Cahn-Hil l iard Equation 90 4.2 Dynamics of an n-layer Metastable Pattern 95 4.3 Properties of the Metastable Dynamics 101 4.3.1 Simplification of the D A E System (4.16) 101 4.3.2 Comparisons of the Internal Layer Dynamics 107 4.3.3 Other Expl ic i t O D E Systems 112 4.4 Simulation of the Entire Coarsening Process 113 5 Numerical Analysis of an Exponentially Ill-Conditioned B V P 124 5.1 Introduction 124 5.1.1 The Analyt ica l Behavior of Solutions 129 5.2 Difference Schemes and their Uniform Convergence 132 5.2.1 Difference Schemes and Some Preliminaries 132 5.2.2 Convergence Analysis 139 5.3 Numerical Experiments and Discussions 142 v 5.3.1 A Model Problem of (5.1) 142 5.3.2 A Nonlinear Problem and a Time-dependent Problem 145 5.3.3 A Spectral Method 152 6 Summary and Future Work 158 6.1 Summary 158 6.2 Future Research 161 Bibliography 163 Appendices 172 A Estimating the weight function u> 172 B Derivation of Equation (3.1) 174 vi List of Tables 2.1 Comparison of asymptotic and numerical values for A 0 with f(u) = u 2 /2 and x0 = 0.50 39 2.2 Comparison of asymptotic and-numerical values for A 0 with f(u) = u 2/2 and XQ — 0.35 39 2.3 Comparison of asymptotic and numerical values for A 0 with f(u) = u — 2 + 4 / 0 + 2) and x0 = 0.40 40 2.4 Comparison of asymptotic and numerical values for Ao with f(u) = u — 2 + 4/(u + 2) and x0 = 0.35 40 2.5 A comparison of the asymptotic and numerical results for the tip t = t(xo) of the flame-front for (2.5) with e = 0.004 and zg = 0.4 46 2.6 A comparison of the asymptotic and numerical results for the tip t = t(xQ) of the flame-front for (2.5) with e = 0.002 and = 0.3 47 2.7 A comparison of the asymptotic and numerical results for t = t(xo) for the asymmetric / ( « ) of (2.80) with e = 0.004 and x° = 0.4 48 2.8 A comparison of the asymptotic and numerical results for t = t(xo) for the asymmetric f(u) of (2.80) with e = 0.003 and = 0.35 49 2.9 The equilibrium location of x™ for the asymmetric f(u) of (2.80) 51 3.1 Example 3.1: A comparison of the asymptotic and numerical results for t = t(x0) 67 3.2a Example 3.2: A comparison of the asymptotic and numerical internal layer trajectories with d = 1.4 and XQ = 0.45 68 vn 3.2b Example 3.2: A comparison of the asymptotic and numerical internal layer trajectories with d = 1.5 and XQ = 0.49 69 3.3a Example 3.3: A comparison of the asymptotic and numerical internal layer trajectories with = 0.4 73 3.3b Example 3.3: A comparison of the asymptotic and numerical internal layer trajectories with XQ = 0.333 74 3.4a Example 3.5: A comparison of the asymptotic and numerical shock layer trajectories with g'(x) = —1.9 and x^ W 0.2057 83 3.4b Example 3.5: A comparison of the asymptotic and numerical shock layer trajectories with g'(x) = —2.1 and X® RJ 0.495 84 3.5a Example 3.5: A comparison of the asymptotic and numerical shock layer trajectories with d = 0.55 and x° ~ 0.2056 86 3.5b Example 3.5: A comparison of the asymptotic and numerical shock layer trajectories with d = 0.45 and x% « 0.495 86 3.6a Example 3.7: A comparison of the asymptotic and numerical shock layer trajectories with x°0 « 0.240 88 3.6b Example 3.7: A comparison of the asymptotic and numerical shock layer trajectories with XQ « 0.250 89 4.1 A comparison of the asymptotic and numerical results for t = t(d3) for the Cahn-Hilliard equation (a = 0) with e = 0.03 101 4.2 A comparison of the asymptotic and numerical results for t — t(d3) for the viscous Cahn-Hilliard equation (a = 0.5) with e = 0.04 102 4.3 A comparison of the asymptotic and numerical results for t = t(d3) for the constrained Allen-Cahn equation (a = 1) with e = 0.04 103 V l l l 4.4 A comparison of the asymptotic and numerical results for ac for the Cahn-Hilliard equation (a = 0) with e = 0.02 106 4.5 A comparison of the asymptotic and numerical results for crc for the viscous Cahn-Hilliard equation (a = |) with e = 0.03 106 4.6 Numerical results for the Cahn-Hilliard equation (a = 0) at two different times 113 5.1 Numerical results of the upwind scheme for (5.52) using double precision. 145 5.2 Numerical results of the coupled scheme for (5.52) using double precision. 146 5.3 Numerical results of the Il'in scheme for (5.52) using double precision. . . 147 5.4 Numerical results of the condition numbers of the coefficient matrices. . . 147 5.5 Numerical results of the three schemes using quadruple precision arithmetic. 148 5.6 Numerical results of the three schemes using single precision arithmetic. . 148 5.7 Numerical results of the scheme (5.56) for the G-L equation using double precision 149 5.8 Numerical results of the scheme (5.56) for the G-L equation using quadru-ple precision 150 5.9 Maximum errors for the spectral method applied to (5.52) 157 ix List of Figures 1.1 The solution to the Burgers equation (1.1) at various times 7 2.1 Diagram illustrating an upward propagating flame in a vertical channel. . 20 2.2 Plot of y(x,t) versus x with e = .0115 obtained from (2.84) 22 2.3 Plots of four equilibrium solutions U(x) versus x for (2.5) with e = 0.005. 24 2.4 Plot of the solution to (2.5) at various times with u(x, 0) = x(l — x)(x — x ° ) . 25 2.5 Plot of the solution to (2.5) at various times with u(x, 0) = —X(1 — X)(X — XQ). 26 2.6 Plots of the solutions to (2.5) at various times with u(x, 0) = x(l — x) (left figure) and u(x,0) = — x(l — x) (right figure) 27 2.7 Plot of the numerical solution to (2.5) at different times 50 2.8 Plots of asymptotic and numerical results for t = t(xo) for (2.5) 51 2.9 Plot of the numerical solution to (2.10) at different times 52 2.10 Plots of asymptotic and numerical results for t — t(xo) for (2.10) 53 2.11 Plots of the equilibrium solutions to (2.11) versus x for various e 53 2.12 Plot of y(x,t) versus x given by (2.84) with e = .006 54 3.1 A cylinder of revolution with cross-section described by R = RoF(X/L). 57 3.2 Example 3.2: Plots of the numerical solutions to (3.1) at different times. 70 3.3 Example 3.3: Plots of the numerical solutions to (3.1) at different times with x°0 = 0.4 and 0.333 71 3.4 Example 3.3: Plots of the numerical solutions to (3.1) at different times with x°0 = 0.76 and 0.79 72 3.5 Example 3.5: Plots of Mi(x0) and M2(x0) 85 x 3.6 Example 3.6: Plots of Mi (x 0 ) and M2{xQ) 87 3.7 Example 3.6: Plots of the numerical solutions to (3.5) at different times. 88 3.8 Example 3.7: Plots of the numerical solutions to (3.5) at different times. 89 4.1 Free energy of the system below the critical temperature. .' 91 4.2 An ra-layer metastable pattern for the viscous Cahn-Hilliard equation. . . 97 4.3 Plot of the numerical solution at various times to (4.11) with a = 0. . . . 100 4.4 Plot of the numerical solution at various times to (4.11) with a = 0.5. . . 104 4.5 Plot of the numerical solution at various times to (4.11) with a = 1. . . . 105 4.6 Plots of the numerical solutions to (4.11) at different times for various values of a and e. Here x° = (-0.8, -0.5, -0.1, 0.2, 0.6, 0.8) 120 4.7 Plots of the numerical solutions to (4.11) at different times for various values of a and e. Here x° = (-0.8, -0.5, -0 .1 , 0.2, 0.6, 0.9) 121 4.8 Plots of the numerical solutions to (4.11) at different times for various values of a and e. Here a;0 = (-0.8, -0.4, -0.1, 0.15, 0.4, 0.7) 122 4.9 Plots of the interface locations as a function of time for (4.11) 123 5.1 The mesh generating function \(t) versus t 134 5.2 Plot of t versus A for (5.58) from the asymptotic approximation and from the full numerical approximation using the upwind scheme 151 5.3 Plot of t versus A for (5.58) from the asymptotic approximation and from the full numerical approximation using the coupled scheme 152 5.4 Plot of t versus A for (5.58) from the asymptotic approximation and from the full numerical approximation using the II'in scheme 153 x i Acknowledgements I would like to express my deepest gratitude to my supervisor, Dr. Michael Ward. His insight into mathematics has opened my mind to a new world and his remarkable ability in applying cutting-edge mathematics to real-world situations has guided me to one of the most interesting topics in the world. In addition, he has always been generous with his time in teaching me how to make a rigorous scientific approach and how to document material and write in English. I am fortunate in having him as a supervisor. I am greatly indebted to Dr. Uri Ascher for his insightful comments and stimulating discussions about numerical computation and to Dr. Jim Varah, Dr. Greif Chen and Dr. Xiao-Wen Chang for their very helpful advice for solving ill-conditioning problems. My thanks also go to Dr. Anthony Peirce and Dr. Brian Seymour for their interest and input in our committee meetings. The help from Anthony, Jim, Michael and Uri was not limited to the science; they also supported my personal goals. I acknowledge the University Graduate Fellowship committee and I.W. Kil lam Trusts for their financial support which made my studies at U B C possible. Finally, very special thanks to my wife Yanping and my daughter Yuan (Amanda). Without their understanding, support and love this thesis would not have been completed. xii Chapter 1 Introduct ion The subject of dynamic metastability has recently generated tremendous interest in the mathematical science community and in several areas of application such as in the phase separation of binary alloys, the propagation of flames, the exit problem of a Brownian particle confined by a finite potential well, etc. Metastable dynamics, which refers to physical processes that involve an extremely slow approach to their equilibrium states, is usually associated with particular types of singularly perturbed parabolic partial differen-tial equations. A problem exhibits metastable behavior when the approach to equilibrium occurs on a time-scale of order 0(e~), where C > 0 and e is the singular perturbation parameter. Examples of such equations include: the Cahn-Hilliard equation and the con-strained Allen-Cahn equation (cf. [24], [22], [91]) modeling the slow phase separation of a binary alloy; a Burger-type equation derived in [85] and [77] for flame-front propagation in a vertical channel; the Kolmogorov's backward equation in the exit problem (cf. [72], [75], [92], [93]); the Gierner Mienhardt equation as an activator-inhibitor model in the mathematical biology (cf. [40], [56]). There are a few common characteristics associated with these singularly perturbed problems exhibiting dynamical metastability. A typical feature is exponential ill-con-ditioning, by which we mean that the spectrum of the eigenvalue problem associated with the linearized equation about the steady state solution contains asymptotically exponentially small eigenvalues. As a result of this exponential ill-conditioning, the time-dependent solution approaches its steady state only over an exponentially long time 1 Chapter 1. Introduction 2 interval. Moreover, the steady state itself is often extremely sensitive to perturbations of the boundary values and the coefficients in the differential operator. The significance of such eigenvalues was first recognized for certain linear two-point problems involving boundary layer resonance in [1], [31], [60], [67], and later in [3], [14], [61], [87], [111], [112] for the Cahn-Hilliard equation, the Allen-Cahn equation and the viscous shock problem. Another consequence of this exponential ill-conditioning is that a straightforward ap-plication of the method of matched asymptotic expansions ( M M A E ) fails to determine the solution uniquely. Specifically, since a conventional M M A E approach is incapable of resolving the exponentially small terms that are significant for ill-conditioned problems, it typically yields an asymptotic approximation with undetermined constants. To overcome this deficiency, various modifications of the M M A E approximations have been proposed to treat asymptotically exponentially ill-conditioned problems. A varia-tional principle [44] and its extensions [114] were postulated and used to calculate the leading order asymptotic solutions for certain linear turning point problems exhibiting the phenomena of boundary layer resonance. Explicit matching of crucial exponentially small terms was implemented to find the shock-layer locations for the steady state Burg-ers equation in [63] and to construct asymptotic expansions of the solutions to several nonlinear boundary value problems in [66]. Another method is applying the nonlinear WKB-type transformation introduced in [86], [87] for the Ginzberg-Landau equation and the viscous shock problem, by which the problems are transformed to well-conditioned steady state problems. In this way, a conventional boundary layer theory or a traditional numerical method can be applied to find the equilibrium solutions. A more powerful tech-nique to resolve this indeterminacy is the projection method that combines the M M A E approach with some information concerning the spectral properties associated with the linearized problem. This method, motivated by the work of deGroen [31], was originally developed by Ward [108] for some reaction-diffusion models and has been successfully Chapter 1. Introduction 3 applied to various problems including the one dimensional exit problem in [67], the vis-cous shock problem in [87] and several phase separation models in [86], [88], [109] and [110]. The projection method has an advantage over other approaches in its adaptability to treat various problems involving indeterminate constants where others may fail to. In addition, using this spectral approach it is possible to derive equations of motion for the internal layers or other structure patterns pertaining to some time-dependent partial differential equations exhibiting metastability. The characterization of these metastable patterns in various physical models in one spatial dimension has become the subject of much recent research. Metastable behav-ior for the time-dependent viscous shock problem was first observed numerically in [61], and a quantitative characterization of this exponentially slow dynamics was derived in [64], [65] and [87] by different approaches. The work in [26], [38] and [39] dealt with the unconstrained Allen-Cahn equation in the one-dimensional case and established the exponentially slow motion of the internal layers. The Cahn-Hilliard equation, model-ing phase separation, has been studied numerically in [76], [32], [9] and the existence of metastable phase field boundaries has been proved in [3], [17], [15], [43] and [35]. An explicit characterization of metastability for the (constrained) Allen-Cahn equation and the (viscous) Cahn-Hilliard equation can be found in [86], [88] and [109], where the asymptotic projection method is used to obtain the equation of motion for the locations of the internal layers. In a multi-dimensional setting, dynamic metastability for phase separation models that conserve mass can also occur and were discussed in [3], [91] and [110]. In spite of numerous efforts devoted to study metastable dynamics in various physical problems in the past decade, there still remain many interesting problems un-explored, especially in multi-dimensional domains and for systems of reaction-diffusion equations. In addition, even for phase separation models in one spatial dimension, quite a few questions and phenomena require further explanation. Chapter 1. Introduction 4 In the rest of this section, we first in §1.1, through studying the Burgers equation for viscous shocks, illustrate some basic characteristics associated with the singularly perturbed problems exhibiting dynamic metastability. Then in §1.2 we outline how the projection method can be used to calculate the undetermined constants occurring in M M A E approximations and to analyze metastable dynamics for time-dependent prob-lems. In §1.3, we propose an algorithm based on the idea of the transverse method of lines (cf. [7]), which will be used to calculate full numerical solutions to the time-dependent singularly perturbed problems studied in this thesis and to compare them with the cor-responding asymptotic approximations. Finally, the contribution of this thesis is given in §1.4. 1.1 Burgers Equat ion for Viscous Shocks We investigate the following initial-boundary value problem for Burgers equation in the limit e —*• 0 : ut + uux = euxx, 0 < x < l , t>0, (1.1a) u(x,0) = u0(x), u(0,t) = a, u(l,t) — —a. (1.1b) Here a is a positive constant. In this equation, the term uux represents a nonlinear convection or transport term, and euxx is a Fickian diffusion term. The nonlinear con-vection term steepens the initial waveform, while the diffusion term attempts to smear out the solution. Thus (1.1) is a balance equation between these two effects. Equation (1.1) was first suggested by Bateman [13] to describe the discontinuous motion of a fluid whose nondimensionalized viscosity e tends to zero. This equation and its generalizations have subsequently been successfully applied in a number of fields, including turbulence, laminar transonic flow, traffic flow, supersonic flow about an airfoil, biochemistry, etc. (cf. [18], [69], [68], [19] [37] and [52]). The ± a boundary values were selected so that a Chapter 1. Introduction 5 traveling wave solution for (1.1a), connecting u = a and u — —a, has zero speed. Con-sequently, the shock layer solution u(x,t) of the initial-boundary value problem (1.1) converges, in time, to its steady state u(x, oo) only over exponentially long time. This sluggishness was attributed by [61] to the occurrence of the asymptotically exponentially small principal eigenvalue for the linearized equation about the steady state solution. We first consider the equilibrium problem of (1.1): £Uxx — uux = 0, 0 < x < 1, (1.2a) u(0) = a, u(l) = - a . (1.2b) This equation has an exact solution u = ue with an interior shock given by ue = — 0t&nh.[0e~1(x — | ) /2] , where 0 satisfies 0 a = /?tanh — . (1.3) 4e Using successive approximation (as in [62]), we can solve (1.3) to obtain 8 = a + 2ae~£ + 0(e~^) , as e -> 0 . (1.4) The eigenvalue problem associated with the linearized equation about ue can be writ-ten as £<f>xx - (ue<f>)x = Ac/>, 0 < x < 1, (1.5a) ^(0) = <j>(\) = 0 . (1.5b) Let AQ denote the first eigenvalue of (1.5). The estimate in [61], based on the Rayleigh quotient, showed that AQ = 0(e~~) for some constant C > 0. More precisely, this eigenvalue has been explicitly calculated in [87] to be AQ ~ -2a2e~1e~ae~112 + as e -> 0 , (1.6) Chapter 1. Introduction 6 which is exponentially small. The exponentially ill-conditioning of the linearized problem suggests that the equilib-rium solution to (1.2) is rather sensitive to the small changes in the boundary conditions and in the coefficients of the equation. In fact, for problem (1.2a) with the following boundary conditions: u(0) = a - Aie-as~1/2 , u(l) = a + Are-QE~1/2 , (1.7) where A\ and AT are positive constants, Reyna and Ward [87] showed that the shock layer solution is given asymptotically by u ~ —a tanh a ( x ~ x ) ; where x* is defined by x-* = ^ + - l o g [ 7 + (7 2 + l ) ^ , 7 = (A - A,)/4a . (1.8) Z a This example shows that the exponentially small changes in the boundary conditions result in an 0(e) change in the location of the shock layer. This property is called "supersensitivity''' in [62], [64], [64] and [65]. There are other examples in these papers showing the substantial perturbation of the shock layer location x* resulting from expo-nentially small changes in the coefficients of the differential equation as well as in the boundary values. As shown in [61], the speed of approach of the solution u(x,t) of (1.1) to its steady state is determined by the principal eigenvalue AQ of (1.5). Since AQ is exponentially small, the solution of (1.1) becomes quasi-stationary and the shock creeps extremely slowly to the equilibrium position once the shock profile has been formed from initial data. Specifically, the dynamical behavior of the solution to the Burgers equation (1.1) can be divided into two different time phases: a transient 0(1) phase where a shock layer of width 0(e), connecting u = a and u — —a, is formed from monotone decreasing initial data; and an exponentially slow phase where the shock layer drifts towards its equilibrium location at an exceeding slow rate. This shock layer is closely approximated Chapter 1. Introduction 7 by the traveling wave solution of (1.1a): ue(x;x0(t)) = - a t a n h "J"^ ) , 0 < x < 1 , (1.9) and its initial location xo(0) depends on the initial data u0(x). In Figure 1.1, we plot the solution to the Burgers equation at various times to show these two different phases. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Figure 1.1: The solution to the Burgers equation (1.1) at various times with e = 0.02 and initial data u(x, 0) = 2x2 — 4x +1. Notice the slow motion of the shock layer towards the equilibrium solution. Using the Cole-Hopf transformation (cf. [30] and [49]), by which Burgers equation can be reduced to the initial-boundary value problem for the linear diffusion equation and thus can be solved analytically in closed form, Reyna and Ward [87] wrote the exact solution to (1.1) in terms of the eigenfunctions of the linear diffusion operator and derived the explicit dynamical behavior for (1.1). Specifically, if the initial data UQ(X) satisfies Chapter 1. Introduction 8 uo(0) = —uo(l) = —a and u'0(x) < 0 for 0 < x < 1, then, for e —> 0, the slow motion of the shock layer is described by (1.9) where xo(t) satisfies with t* = e(2a2)~1 e £ and 4 = l + ^~ [1uo(x)dx + ^[u'o(l)-u'o(0)} + . . . , as e - 0. (1.11) It is clear from (1.10) that it will take 0(t*) time scale for the shock layer to reach its equilibrium position at x0 = | . Finally, we note that, for e —• 0, the leading order M M A E solution to (1.2) is given by it ~ us(x;xo), where the shock profile ue(x;xo) satisfies (1.9) and the shock location is an undetermined parameter. It is obvious that for any x0 £ (0,1), with x0/e ^> 1 and (1 — xo)/e 3> 1, this leading order M M A E solution satisfies (1.2a) exactly and (1.2b) to within exponentially small terms. Therefore, we conclude that the correct value .To = | can not be determined analytically even after calculating higher order boundary layer corrections near each endpoint. This indeterminacy in xo can be eliminated by symmetry considerations, by constructing a higher order M M A E solution accounting for exponentially small terms (cf. [62]) or by the projection method, which we will introduce below. 1.2 The Pro jec t ion M e t h o d The idea of the projection method was originally introduced by Ward [108] to determine the locations of the internal layer positions for certain nonlinear singularly perturbed boundary value problems. It was later generalized by Ward and his co-workers in [67], [79], [86], [87], [88], [89], [110], [111], [112] to study the metastable dynamics for various classes of time-dependent singularly perturbed problems. The basic idea behind this Chapter 1. Introduction 9 method is to supplement the method of matched asymptotic expansions with certain spectral information associated with the linearized equation. At the outset we illustrate the key elements of the projection method by studying the following system of linear equations Asx = b (1.12) where x and b are dimensional vectors and A , depending on a small parameter e > 0, is a n J V x A f Hermitian matrix with eigenvalues A a < A 2 < . . . < XN and corresponding orthonormal eigenvectors <£x, <f)2,..., 4>N- In addition, we assume that the right-hand side b depends on a vector of unknowns, say a <E RM with m < N. Now, let's determine this unknown vector a by assuming some properties of A specified below. Suppose that A E does not have a zero eigenvalue. Then we can expand the solution x in terms of eigenvectors of A E : z = £ 4 ^ - , (1.13) 3 = 1  A3 where (b,<j)j) = bT<$>3 . We further assume that the first m eigenvalues A i , . . . , A m are exponentially small as e —> 0 so that A, ~ 0 ( £ p e - T ) , j = l , . . . , m , (1.14) where C > 0 and p are constants independent of e, and A T O + 1 , . . . , Ajv are bounded away from zero as e —> 0. Then, for a solution of (1.12) to exist in the limit e —> 0, we require that as e —> 0, ( & , & ) - * ( ) , for j = l , . . . , m . (1.15) Setting (b,4>j) = 0, for j = l , . . . , m , we obtain a system of algebraic equations for determining the unknown vector ct € RM. Chapter 1. Introduction 10 We now apply this technique to resolve the indeterminacy occurring in the conven-tional M M A E approximations and to derive equations of motion for the shock-like layers or other localized structures pertaining to some evolution equations exhibiting dynamic metastability. Consider the following steady state nonlinear differential equation of the form NE(u) = euxx + F(u,ux) = 0, 0 < x < 1, (1.16a) Biu = eux — KI(U — uo) — 0 at x — 0 , (1.16b) BTu = eux + Kr(u — u\) = 0 at a; = 1 . (1.16c) Here e —> 0 is a small parameter, no, ui, Ki > 0 and nr > 0 are constants, and F(- , •) is a nonlinear function. Assume that we can find a M M A E solution uE[x; a i , . . . , ctm] to (1.16), which represents an m-parameter family of "approximate" solutions with integer m > 1 and parameter ctj belonging to some set Sj, so that for each ctj G Sj, j = 1 , . . . , m, we have Ne{ue) = 0{epe-*) , (1.17a) Biu£ = O( £ p o e-T-) , Bru£ = 0{ePle~^) , (1.17b) where p, p0, pi, q > 0, <?o > 0 and qi > 0 are constants. Then the conventional M M A E does not give a uniquely determined approximation, unless exponentially small terms are taken into account. To select the correct vector a= ( a i , . . . ,ctm)T corresponding to a true steady state solution, we linearize (1.16) around u€ by writing u = u£ + v to get L£v = evxx + F2(u£, u\)vx + Fi{ue,uEe)v = -NE(us) ,0<x<l, (1.18a) Biv = —Bius at x = 0 , Brv = —BTue at x = 1 , (1.18b) where Fi(u,v) = and F2(u,v) = dFQ^• Noting that the differential operator L £ with homogeneous boundary conditions can always be transformed into a self adjoint Chapter 1. Introduction 11 form using a Liouville transformation in some weighted space UJ, we let X3 and (j>j for j > 1 be the normalized eigenpairs in this space of the associated eigenvalue problem Le(f) = \(l>, 0 < a r < l , £ , u ( 0 ) = 0 , Bru(l) = 0. (1.19) Then we write the solution v to (1.18) in terms of the eigenfunctions (j>j as v = ^Zj^i v <A? • Here the coefficient Cj can be obtained from (1.18) and the Lagrange's identity as Cj- = -(Ne(ue), tj>j)w + B.T. (boundary terms) , (1.20) where the inner product is defined by (u,v)w = JQ uvuidx and B.T. denotes some items depending on the values and/or derivatives of us, cf>j and UJ at the endpoints. Let (f>j = -^-u£[x; c*i,... ,cxm]. Then, by differentiating (1.17) with respect to <XJ, we get Le<j)j — e.s.t. (exponentially small terms) , 0 < x < 1 , (1.21a) E>i<f)j — e.s.t. at x — 0, Br<j)j = e.s.t. at x = 1. (1.21b) Thus, if these functions (f>3•, j = l , . . . , m , are independent, then (1.21) suggests that (1.19) has m eigenvalues that are exponentially small. Since Ls(j>j is uniformly exponen-tially small, we have that <j)j is proportional to </>j, except near the endpoints at x = 0,1 where boundary layer corrections must be inserted in order to satisfy the boundary con-ditions. When the asymptotic approximations of the eigenfunctions <f>j , j = 1, . . . , m, are obtained, we can estimate the eigenvalues Xj, j' = 1, . . . , m . If these eigenvalues are shown to be exponentially small and other eigenvalues A m + i , . . . ,A ;v are negative and bounded away from zero as e —>• 0, then a necessary condition for the solvability of (1.16) is that Cj• —> 0 as e —• 0, j = 1, . . . , m. Setting Cj = 0 in (1.20), we obtain a system of nonlinear algebraic equations for determining the unknown parameter vector oc (Ns(iie),<f>j)w = B.T. , j = l , . . . , m . (1.22) Chapter 1. Introduction 12 We now outline how the projection method can be used to analyze metastability for the time-dependent problem corresponding to (1.16): ut = euxx + F(u,ux), 0 < x < 1, (1.23a) BlU{0) = 0, £ r u ( l ) = 0 ; u(x,0) = u0(x). (1.23b) We seek a solution to (1.23) for t ^ > 1 in the form u(x,t) = ^[sjai(<),...,am(<)] + u ( M ) , (1.24) where i; « ? and i>t <C dtue • Linearizing (1.23) around u e , we obtain that v satisfies the quasi-steady problem m Lev = -Ne(ue) + Y,<x'jdajue, 0<x<l, (1.25a) i=i B/u = - 5 , u £ at x = 0 , B r u = -Brue at x = 1. (1.25b) Here a'j = da3(t)/dt and the operator Ls is the same as in (1.18). Applying the similar technique used in studying the equilibrium problem (1.16), we expand v in terms of as v = z l j ^ i ^y^^j > where Cj(<) = - ( y Y e ( £ i e ) , ^ ) u , + B.T. + £ X ( 0 Q i u e , & ) , j = l , . . . , m . (1.26) t=i Since the eigenvalues Xj,j = l , . . . , m are assumed to be exponentially small, for a solution of (1.23) to exist in the limit e —* 0, we require that Cj —• 0 as £ —> 0. Then, by letting c, = 0 in (1.26), we get m (Ne(ue),<f>j)„ = B.T. + S ^ i f i e ) ^ ) I j = l , . . . , m . (1.27) i=l This is a system of ordinary differential equations (ODEs), from which the dynamics of m parameters cxi(t),..., am(t) can be derived. Chapter 1. Introduction 13 In summary, the metastable dynamics for (1.23) is characterized by u ~ us[x;ai(t), ..., am(t)], where the quasi-equilibrium ue is the M M A E solution and otj(t), j = 1, . . . , m is determined by the system of ODEs (1.27). The equilibrium values a] for ctj(t), j = 1, . . . , m, corresponding to the equilibrium solution for u(x,t), satisfy (1.22) which can also be obtained by setting a'3 — 0 in (1.27). 1 . 3 T h e N u m e r i c a l M e t h o d T M O L To verify the ODE system (1.27) and other asymptotic results in the thesis numerically, we need to compute the numerical solutions to the time-dependent singular perturbation problems directly. The numerical algorithm we will use is called the transverse method of lines (TMOL) (cf. [7]) which is applicable to various types of parabolic partial differential equations or systems, especially in one spatial dimension. For illustration purposes, we use the T M O L approach to compute numerical solutions to (1.23). The T M O L is based on replacing the time derivative in (1.23) by a difference ap-proximation and then solving the resulting boundary value problems in space. More specifically, suppose tj, for j — 0,1,.., are the grid points in time that are determined in the actual computation using a time-stepping control strategy. Then, we convert the time-dependent problem (1.23) to a set of boundary value problems using the Backward Differentiation Formulas (BDF) (cf. [7]) k 5>j"n-j(:c) = 0n+1N£un(x), B,un{0) = Brun{l) = 0 . (1.28) i=o Here 0O = 1 and the differential operator Ne is defined in (1.16a). The other coefficients 0j, for j > 0, which depend only on hi = U — for i = n, n — 1,.., n — k + 1, can be computed numerically using Gaussian elimination in such a way that the B D F scheme (1.28) is k-th. order accurate in time. Chapter 1. Introduction 14 For every fixed n, (1.28) is a two-point boundary value problem which we solve using COLSYS ([6]). Although this approach is computationally expensive, it yields approximate solutions to (1.23) that are highly accurate in space. Since several time scales may occur in our problem, we found it necessary to implement a time-stepping control strategy to efficiently track the solution to (1.1) over long time intervals. To achieve this, we employed a higher (e. g. , (k + 1)—th) order B D F scheme at each time step for the purpose of comparison, and used the /2-norm of the difference between the solutions of the k-th and the (k + 1)—th. order B D F schemes as an error indicator to reject large inaccurate time steps or to enlarge unnecessary small time steps. In all of the calculations below we took k = 2. Comparing the T M O L with the method of lines (MOL) which discretizes a time-dependent P D E in space first and then solves the resulting initial value problem (IVP) in time, we found that the T M O L is easier to implement and is rather accurate for our metastable problems. The numerical computations of various metastable dynamics showed that the global numerical errors are strongly dependent on the discretization er-rors in spatial variable, whereas they are usually less sensitive to the time discretization. Thus, we consider that a good implementation of a spatial discretization is critically significant for a singular perturbation problem exhibiting metastable dynamics, whose solutions often possess kinds of singularities in space such as internal layers and boundary layers. Fortunately, COLSYS is a well-known effective software to yield a numerical solu-tion adaptively to a boundary value problem within a prescribed precision. Incorporating COLSYS into the T M O L makes it easier to treat various parabolic singular perturba-tion problems in this thesis. Since our time discretization (2nd order B D F scheme) does not depend on the specific nonlinearity, the major work to compute a new problem is to define the parameters and supply the subroutines required by COLSYS to solve the semi-discretized boundary value problems such as (1.28). On the other hand, if the Chapter 1. Introduction 15 M O L is employed to solve the time-dependent PDEs, we will have to pay much attention to constructing the scheme and designing the corresponding mesh individually for each problem considered. This method has another obvious drawback in that the spatial mesh does not change. This means that for our metastable problems having transient regions like a moving shock layer, a fine mesh should be used throughout the whole interval in space. In other words, we do not have the flexibility in treating the spatial variation un-less certain moving mesh techniques (cf. [54]) are employed. Moreover, even for problem without internal layer regions, the M O L has a severe limitation of the number of mesh points in space clue to the capacity of a computer, if the resulting IVP is to be solved by an IVP software not accounting for the sparse structure of the right-hand side of the IVP. In consequence, we can not guarantee that the numerical results of the M O L are accurate enough to examine the validity of our asymptotic results. 1.4 C o n t r i b u t i o n of This Thesis The first goal of the thesis is to further the development and application of the projection method to certain time-dependent singular perturbation problems having metastable dynamics. The problems included in this thesis research fall into three categories: an upwardly propagating flame front in a vertical channel, internal layers in a weakly varying geometry and some phase separation models. In Chapter 2, we study a Burgers-type equation modeling an upward flame front propagation in a vertical channel. For this problem, it is shown that the principal eigen-value associated with the linearization around an equilibrium solution corresponding to a parabolic-shaped flame-front interface is exponentially small. This exponentially small eigenvalue then leads to a metastable behavior for the time-dependent problem. This Chapter 1. Introduction 16 behavior is studied quantitatively by deriving an asymptotic ordinary differential equa-tion characterizing the slow motion of the tip location of a parabolic-shaped interface. The asymptotic results complement the rigorous, but qualitative, metastability result obtained in [16]. Similar metastability results are obtained for a more generalized Burg-ers equation. These asymptotic results are shown to compare very favorably with full numerical computations. Most parts of this chapter are taken from the paper [100]. In Chapter 3, the projection method is applied to study two time-dependent singularly perturbed problems related to exponentially slowly varying geometries. The first problem is a Burgers-like convection-diffusion equation which describes one dimensional transonic flow through a nozzle with a weakly variable cross-sectional area. The metastable be-havior of the shock waves occurring in the nozzle is studied quantitatively by deriving an asymptotic ODE characterizing the slow motion of the shock layer. From this ODE, we found that a stable steady shock layer may exist in the convergent part of a nozzle, which seems to contradict the previous experimental and analytical result that stable steady shock layers only occur in the divergent parts of a nozzle. The disagreement is explained. The second problem we consider in this chapter is a generalized Ginzburg-Landau(G-L) equation in one dimension, which is employed to determine conditions for the existence of stable spatially-dependent steady state solutions to the Ginzburg Lan-dau equation in several space dimensions. In a convex domain, the Ginzburg-Landau equation ut = s2Au + Q(u) with Neumann boundary conditions does not admit stable spatially-dependent steady state solutions. However, this result does not hold for non-convex domains. From the ODE describing the metastable dynamics of this generalized G-L equation, which arises from an asymptotic reduction of a G-L equation in a long, thin, axially symmetric channel, we show that non-constant stable steady solutions to the G-L equation may exist in some non-convex domains. Most parts of this chapter are taken from the paper [101]. Chapter 1. Introduction 17 In Chapter 4, we consider a viscous Cahn-Hilliard equation in one spatial dimension, from which several phase separation models, including the constrained Allen-Cahn equa-tion, the viscous Cahn-Hilliard equation and the Cahn-Hilliard equation, can be derived by letting a continuation coefficient take on some limiting values. The metastable be-havior associated with these phase separation models is quantitatively described by an asymptotic system of differential algebraic equations (DAEs) derived by applying the projection method. Through simplifying this system of DAEs, we identify the differences and similarities of the metastable behavior associated with the various phase separa-tion models and we compare our asymptotic results with some previous results for the metastable dynamics associated with the Cahn-Hilliard equation. Our analysis is verified numerically by solving the original viscous Cahn-Hilliard equation and the asymptotic system of DAEs directly. In addition, a hybrid algorithm based on the asymptotic infor-mation and the conservation of mass condition is proposed to model the entire coarsening process associated with the phase separation models. Most parts of this chapter are taken from the paper [102]. From a numerical point of view, it seems that there is an insurmountable obstacle in solving exponentially ill-conditioned singular perturbation problems, say Leu = f. More specifically, since u(x) can be exponentially sensitive to all the data in the equation, e.g., f(x), a perturbation A / in f(x) may cause rather large changes in u(x), of the order 0(\Q 1 A / ) , where Ao (exponentially small) is the principal eigenvalue associated with the linearization operator of Ls. Thus, one may naturally guess that the numerical truncation error must be less than the order of the principal eigenvalue Ao to guarantee the conver-gence of the numerical method. Based on this conjecture, it seems impractical to treat exponentially ill-conditioned problems by using conventional numerical methods, since A 0 might be smaller than the machine precision. On the other hand, however, we have noticed that many classical numerical schemes with moderate size meshes have been Chapter 1. Introduction 18 successfully applied to obtain approximate solutions to these ill-conditioned problems (cf. [32], [87], [88], [9], [8], [62]). Therefore, the second goal of the thesis is to explain why these classical schemes succeed and shed some light on numerical computation of expo-nentially ill-conditioned problems by studying the "simplest" linear metastable problem — a boundary layer resonance problem in Chapter 5. For this resonance problem, bounds for numerical errors for the upwind scheme (cf. [90]), the coupled scheme (cf. [104]) and the Il'in scheme (cf. [5], [55]) are established rigorously. These bounds demonstrate our observations from numerical experiments that although the discrete stability estimate is not valid for a numerical scheme of an expo-nentially ill-conditioned problem, a truncation error may not result in very large errors in the numerical solution and a scheme may still be uniformly convergent with respect to e (i.e., the convergence constant does not depend on e). However, the coefficient matrix of a scheme will usually inherit the ill-conditioning from its continuous problem and con-sequently, we have to resort to high precision arithmetic to yield a true solution to the scheme. In addition, through solving the time-dependent problem corresponding to the boundary value resonance problem and a steady Allen-Cahn equation numerically, we further believe that the traditional finite difference schemes and other standard numer-ical methods may also provide accurate approximation solutions to the time-dependent metastable problems and to nonlinear problems exhibiting exponentially ill-conditioning. Finally, a summary of the major results in this thesis and the potential problems for future research are presented in Chapter 6. Chapter 2 Metastability in Upward Propagating Flame 2.1 A Generalized Burgers Equation We now apply the projection method to study the dynamics of an upward propagating flame-front in a vertical channel modeled by a generalized Burgers equation. We begin with an outline of the physical background of this equation. There are three basic distinct types of phenomena that may be responsible for in-trinsic instabilities of premixed flames: body-force effects, hydrodynamic effects and diffusive-thermal effects. Within the framework of a one-dimensional slab geometry (see Figure 2.1), the weakly nonlinear flame interface evolution equation describing the effects of buoyancy under conditions of diffusive-thermal instability of the flame was proposed by Rakib and Sivashinsky [85] as Ft - \ubF2x - aDthFxx - 4Dthl2hFxxxx = ^r(F- < F >) . (2.1) l ZUb Here y = F(x,t) is the perturbation of a planar flame front y = Ubt; < F > is the space average over the gap between the vertical walls x = 0 and x = L\ Ub is the flame speed related to the burnt gas; g is the acceleration of gravity; 7 = (/>_ — p+)/p_ is the thermal expansion coefficient of the gas; p_ and p+ are the densities of the unburnt and burnt gases, respectively; Dth is the thermal diffusivity of the gaseous mixture; lth is the thermal thickness of the flames; a — \&(l — Le) — 1 where 0 is the Zeldovich number and the Lewis number Le = Dth/Dmoi is the ratio of the thermal diffusivity of the mixture to the molecular diffusivity of the deficient reactant. 19 Chapter 2. Metastability in Upward Propagating Flame 20 Figure 2.1: Diagram illustrating an upward propagating flame in a vertical channel. Equation (2.1) covers the whole range of buoyancy and diffusive-thermal effects in the flame and is a rigorous asymptotic equation derived from some equations of aerothermo-chemistry (cf. [85], [96]). In the absence of the effects for buoyancy, the asymptotic equa-tion (2.1) will reduce to the well known Kuramoto-Sivashinsky equation, i.e., (2.1) with a homogeneous right hand side. In a particular parameter region (47r(2 — "f)U2/"fgL >^ 1) (cf. [77]), a nonlinear analysis in [96] shows that one can also consider the equation Ft - \uhFZ = D instead of (2.1). This equation was derived within the framework of the Boussinesq model which neglects density variation everywhere except in the external forcing term. Here we assume that the Lewis number Le is large enough to ensure the positive sign of the coefficient of the second derivative. Otherwise, the corresponding evolution problem is i l l -posed. Assuming that the walls are thermally insulating, we thus consider the evolution b(l-Le)-l Fxx + ljL(F- < F >) , (2.2) Chapter 2. Metastability in Upward Propagating Flame 21 equation (2.2) subject to the adiabatic boundary conditions Fx(0,t) = Fx(l,t) = Q. (2.3) In terms of certain appropriate dimensionless variables, problems (2.2), (2.3) may be written as 1 f1 Vt ~ TyVl =  eVxx + y - J ydx, 0<x <l, t> 0, (2.4a) yx(0,t) = 0 , yx(l,t)=0; y(x,0) = yo(x). (2.4b) Here e = 2aDthUb/"/gL 2 > 0 is a small parameter and the dimensionless variables t and x have been chosen to share the same notations for the corresponding dimensional vari-ables for convenience. Finally, the substitution u(x,t) = —yx(x,t) leads to the following Burgers type equation ut -f- uux — u = euxx , 0 < a; < 1, t > 0 , (2.5a) u(0,t) = u ( l , £ ) = 0 , u(x,0) = uo(x). (2.5b) As shown numerically in [77], the solution to (2.4) (or (2.5)), for a certain class of in i t ia l conditions relevant to flame-front propagation, exhibits a phenomenon known as dynamic metastability when e <C 1. In Figure 2.2 we illustrate this metastable behavior by plott ing some numerical results for the shape of the interface y = y(x,t) versus x at four different values of t when e = 0.0115. In Figure l a we choose an in i t ia l condition where the flame-front assumes a somewhat concave parabolic shape. Then, as shown in Figure 2.2a-c, the tip location XQ = xo(t) of the parabola, defined as the location of the m a x i m u m value of y at time t, moves towards the channel wall at x = 0 rather slowly. For other ini t ia l conditions, the tip of this interface can move slowly towards the other wall at x — 1. When e is decreased, this stage of the motion, whereby the tip of the Chapter 2. Metastability in Upward Propagating Flame 22 parabolic flame-front moves towards one of the walls, becomes exceedingly slower than in Figure 2.2a-c. In [16] it was proved that this motion is asymptotically exponentially slow as e —> 0. Finally, when the tip of the interface comes close enough to the wall, the rate of evolution of the flame-front increases and a final equilibrium state is attained when the tip touches the wall (see Figure 2.2c-d). Figure 2.2: Plot of y(x, t) versus x with e = .0115 obtained from (2.84). (a) ini-tial quasi-equilibrium solution y(x,t) with tip location x0 = 0.45 at t = 0; (b) quasi-equilibrium solution with XQ = 0.4 at t — 90.05; (c) quasi-equilibrium solution with XQ = 0.3 at t = 113.69; (d) final stable equilibrium solution at t > 117.07. For the equilibrium problem, it was shown in [16] that (2.5) admits multiple equi-librium solutions when e <C 1. In particular, there exists a unique positive equilibrium and a unique negative equilibrium U~. These solutions were found to be linearly stable. In addition, for e <C 1, it was shown that (2.5) has two unstable equilibrium solutions L7 £ +J and U~i, which each have exactly one zero-crossing in the interval (0,1). Chapter 2. Metastability in Upward Propagating Flame 23 Other equilibrium solutions with more than one zero-crossing are also possible. The sta-bility of these equilibrium solutions with more than one zero-crossing and the associated time-dependent solutions were studied in [41]. In Figure 2.3 we plot the four equilib-rium solutions £/"+, U~, U*x and U~x when e = 0.005. From this figure we observe that and U~ have boundary layers of width 0(e) near one of the endpoints, U*yl has an internal layer of width 0(e) near x = 1/2, and U~x has an 0(e) boundary layer near each endpoint. Among these solutions, U~x corresponds to a concave parabolic-shaped equilibrium flame-front interface. We show that the linearization of (2.5) around U~x has an exponentially small principal eigenvalue as e —> 0. Thus, it is this equilibrium solution that is the most significant for the occurence of metastable behavior for the time-dependent problem. For the time-dependent problem, our numerical computations and the results in [77] and [16] suggest that the occurrence of metastable behavior for (2.5) strongly depends on the initial condition. In particular, from [16], a sufficient condition as e —> 0 for metastable flame-front dynamics for (2.5) (or equivalently (2.4)) is that the initial data UQ(X) satisfies UQ(X) < 0 for x € (0,a), and UQ(X) > 0 for x £ (a, 1), (2.6) where a > 0. For other cases, our numerical computations suggest that a stable equilib-rium configuration can usually be attained in an 0(1) time interval. In Figure 2.4-2.6 we illustrate the dynamics of the solution u to (2.5) for various initial conditions. Only in Figure 2.4, where the initial data satisfies (2.6), is an exponentially slow motion observed. Therefore, when the initial data satisfies (2.6) and when e <C 1, we have three different time behaviors under (2.4): a transient 0(1) phase where the parabolic-shaped flame-front interface is formed; an exponentially slow phase where the tip of the parabolic Chapter 2. Metastability in Upward Propagating Flame 24 1 0.8 0.6 0.4 0.2 ^ 0 -0.2 -0.4 -0.6 -0.8 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Figure 2.3: Plots of four equilibrium solutions U(x) versus x for (2.5) with e = 0.005. The solid line and the dashed line represent the positive equilibrium solution and the negative equilibrium solution U~, respectively. The dotted line and the dash-dotted line show two unstable equilibrium solutions U~x and U^. The solution U~x is closely related to the metastable parabolic-shaped flame front. flame-front drifts towards one of the walls; an 0(1) collapse phase where the flame-front collapses against the wall and attains its equilibrium configuration. In terms of u(x, t), the first two phases are clearly seen in Figure 2.4. The fact that the time interval corresponding to the second phase may become exceptionally long when e is small creates an illusion that the flame-front has reached some final equilibrium. However, this phase is merely a quasi-equilibrium transient phase that persists for an exponentially long time interval. If the initial data has several interior zeros, then we would expect that shock layers would quickly be formed with each shock layer connecting two adjacent segments Chapter 2. Metastability in Upward Propagating Flame 25 where u is roughly linear in x . Numerical computations, which we do not give, show that in certain cases metastable patterns may happen. 0-61 1 1 1 1 , — -, , , , 0.5 0.4 0.3 0.2 S 0.1 0 -0.1 -0.2 -0.3 -0.4 — t=0 - t=1.48 - - t=3.08 t=2.147e8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x 0.9 Figure 2.4: Plot of the solution to (2.5) at various times with e = 0.003, XQ = 0.45, and initial data u(x, 0) = x(l — x)(x — XQ). Notice that the zero of u, which is the tip of the parabolic flame-front interface, moves slowly towards the wall at x = 0. One of our objectives is to use the projection method to give an explicit asymptotic characterization of metastable flame-front motion for (2.4), (2.5) in the limit e —> 0. The asymptotic results complement the rigorous, but qualitative, metastability result obtained in [16]. Using the method of matched asymptotic expansions, a quasi-steady concave parabolic-shaped flame-front interface for (2.4) can be expressed in terms of u(x,t) as u(x,t) ~ us[x; Xo(t)], where ue[x; x0] = x — x0 + ui £ X'^ XQ + ur e 1(1 - x); xQ (2.7) Here ui(y; x0) and ur(y; x0) are boundary layer functions that tend to zero exponentially Chapter 2. Metastability in Upward Propagating Flame 26 Figure 2.5: Plot of the solution to (2.5) at various times with e = 0.003, x% = 0.48, and initial data u(x,0) = —ai(l — x)(x — XQ). A shock-layer is formed in an 0(1) time interval and no metastable behavior is observed. as y —> o o , and the unknown XQ satisfies Xo € (0,1). Thus, to within exponentially small terms, Xo is the zero of u. Since yx = —u, it follows that Xo = Xo(t) also represents the trajectory of the tip of the parabolic-shaped flame-front interface for (2.4). For a fixed xQ satisfying x 0 € (0,1), we show below that the principal eigenvalue associated with the linearization of (2.5) around u£ is exponentially small and has the asymptotic estimate Ao ~ \ [x0 (X0 - ce"2) e - S / * + (1 - x 0) ((1 - x 0) - ce1'2) e ~ ^ 2 ^ } , (2.8) as £ —> 0, where c = (S/V) 1/ 2 . This eigenvalue is responsible for the metastable behavior. For the time-dependent problem, we use the projection method to derive an asymp-totic ordinary differential equation for xo(t), which explicitly characterizes the metastable Chapter 2. Metastability in Upward Propagating Flame 27 0.1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 0.1 0 . 2 0 . 3 0 . 4 0 . 5 Figure 2.6: Plots of the solutions to (2.5) at various times with e = 0.01 and initial data i t(x,0) = x ( l - x), (left figure) and u(x,0) = - x ( l - x), (right figure). Metastable behavior does not occur for these initial data. flame-front motion. This method is based on a quasi-steady linearization of (2.5) around u s given in (2.7). Since A 0 is exponentially small, a limiting solvability condition must hold in the limit e —• 0 for the linearized problem. From this condition, we will derive that xo(t) satisfies x n ~ '(1 " * o ) 2 + )^ e - d - o ) 2 / * _ {4 + ^ ) e - . S / 2 . (2.9) Rather than focusing exclusively on (2.5), we instead analyze the following generalized form of (2.5): ut + f(u)ux - f'(u) = euxx, 0 < x < 1, t > 0, (2.10a) u(0,t) = u(l,t) = 0; u(x,0) = u0(x). (2.10b) Here f(u) is smooth, convex, and satisfies /(0) = f'(0) = 0. The special case f(u) = u2/2 yields (2.5). This generalized problem exhibits a very similar metastable behavior as that for (2.5) and is no more difficult to analyze. The chapter is organized as follows. In §2.2, we construct an asymptotic expansion for a certain equilibrium solution of (2.10) and we outline an application of the projection Chapter 2. Metastability in Upward Propagating Flame 28 method to (2.10). In §2.3 we derive a two-term asymptotic expansion for the principal eigenvalue A 0 associated with the corresponding linearized problem and we compare this expansion with full numerical results for Ao- In §2.4, the asymptotic projection method is used to derive an ordinary differential equation characterizing the metastability in (2.10). The results are then applied to the special case f(u) = u2/2, and we show how to recover the flame-front interface y(x,t) satisfying (2.4). In §2.5, we use a full numerical method to compute metastable behavior for (2.10). The numerical results are found to compare very favorably with the corresponding asymptotic results in §2.4. 2.2 T h e E q u i l i b r i u m P r o b l e m We first consider the equilibrium problem for (2.10) in the limit e —» 0 e u M - / ' ( u ) u I + / ' ( « ) = 0, 0< . T <1 , (2.11a) u(0) = 0, u(l) = 0. (2.11b) Here f(u) is smooth, convex, and satisfies /(0) = /'(0) = 0. This problem admits multiple equilibria. However, we will only construct a solution to (2.11) having the form given in (2.7), since it is this solution that is closely related to metastable behavior in the corresponding time-dependent problem. The outer approximation for this solution is clearly u ~ x — Xo for some Xo £ (0,1). This outer solution satisfies the differential equation (2.11a) exactly, but not the boundary conditions (2.11b). Therefore, there are boundary layers near the end points x = 0 and x = 1. In the boundary layer near x = 0 we let y = e~lx and ui(y) = u(ey), and we expand My) = ~xo +  ui0{y) + 4y +  uh{y)} + ••• • (2.12) Substituting (2.12) into (2.11), collecting powers of e, and matching to the outer solution, Chapter 2. Metastability in Upward Propagating Flame 29 we obtain and <-/ ' ( - * (> + = 0 , 0 < y < o o , (2.13a) u/0(°) = , ui0(y) ~ aloe~Uiy as y - * oo, (2.13b) uh ~ lf\-xo + ulo)uh]' = yf"(-x0 + ulo)u'lo , 0 < y < oo , (2.14a) 1^(0) = 0 , uh(y) ~ ahy2e-">y + bhyeT^ as y oo . (2.14b) Here ah - f"(-xQ)alo/2 and 6 ; i = 2<njvi. Upon integrating (2.13), we find that the positive constants ui and a;0 are given by -f'(-xo), f° l o g « ; 0 = l o g .To ~ Ul / (2.15a) ds. (2.15b) .M-f(-xo) f'(-x0)(s + x0) The asymptotic form in (2.14b) is obtained from (2.14a) by using the far-field behavior of u!o(y) as y -»• oo. By integrating (2.13) we obtain the equivalent first order equat U'l0 = / K - X0) - / ( - X Q ) , U/o(0) = X 0 , W i t h U{o(0) = - / ( -cion (2.16) Similarly, in the boundary layer near x = 1 we let y = e 1(1 —x) and ur(y) = u(l—ey), and we expand ur(y) = 1 - x 0 + uro(y) + e[uTl(y) - y] + • • • . (2.17) From (2.17) and (2.11) we obtain the boundary layer equation <0 + f'(l-x0 + uro)u'ro=0, 0 < y < (2.18a) UrQ(0) = XQ - 1 < 0 , "r0 (y) ~ -a r o e V t V as y —> oo (2.18b) Chapter 2. Metastability in Upward Propagating Flame 30 A similar equation holds for un. Here ar0 and vr are defined by vr = f'(l-x0), loga r o = log(l - Xo) + vr I Jo 1-370 + f(l-Xo)-f(3) f'(l-x0)(s-l + x0)\ (2.19a) ds. (2.19b) Consider the left boundary layer expansion (2.12). Using the asymptotic behavior of ui0 and ui1 as y —»• oo, we observe that (2.12) becomes disordered (i.e., euil <C ui0 does not hold) as y —> oo when sy2 — 0(1) or, equivalently, when x = 0(e1/2). Thus, (2.12) holds on the interval y = 0(eq), where — | < q < 0. A similar comment can be made for the boundary layer expansion (2.17) near x — 1. This observation is used in §2.3 and §2.4 to help evaluate certain integrals asymptotically. A composite expansion for this equilibrium solution, valid for x € [0,1], is obtained by combining (2.12) and (2.17). This yields, U ~ U6[x; XQ) = X - XQ + U; 0[£ _ 1X; XQ] + U r o [ £ _ 1 ( l - x); XQ] -\ . (2.20) Here ui0 and UTQ satisfy (2.13) and (2.18), respectively. In (2.20), we have emphasized the parametric dependence of UIQ, UTQ and u£ on the unknown constant x0, which satisfies 0 < xo < 1. When f(u) = u2/2, this constant represents the tip of the equilibrium parabolic-shaped flame-front interface. The difficulty in analytically determining the correct value for x 0 still persists even after calculating higher order boundary layer corrections near each endpoint. By symmetry, when / ( « ) is even, the correct value is clearly Xo = 1/2. However, to determine x 0 analytically for general f(u) we must retain exponentially small terms in the asymptotic expansion of the solution. One way to do this is to use the projection method as shown below. Chapter 2. Metastability in Upward Propagating Flame 31 2.2.1 A n A p p l i c a t i o n of the Pro ject ion M e t h o d We now outline how this method can be used to determine the equilibrium value for .x-o in (2.20) and to analyze metastability for the time-dependent problem (2.10). To analyze metastable behavior for (2.10), we seek a solution to (2.10) for t >^ 1 in the form u{x,t) = ue[x;x0{t)] + v{x,t), (2.21) where v <C ue and vt <C dtu£. Linearizing (2.10) around ue, we obtain that v satisfies the quasi-steady problem Lev = -R + x'0dXou£, 0 < x < 1, (2.22a) v(0,t) = -ir(0;xo), v(l,t) = -ue(l;x0). (2.22b) Here the operator L t and the residual R — R(x; XQ) are defined by Lev = evxx-[f'(ir)v}x + vf"(u£), (2.23a) R = eu£xx - /'(«•)«« + /'(u£). (2.23b) Now consider the homogeneous operator Le where XQ is a parameter. Let Xj, qbj for j > 0 be the normalized eigenpairs of the associated eigenvalue problem Le<j> = \<j>, 0 < * < 1 ; 0(0) = (^1) = 0. (2.24) The Xj are real and the <f>j satisfy the orthogonality relations ( & , ^ ) w = fyfc, = 0,1, (2-25) Here the inner product is defined by (u,v) = f0luvudx, where the weight function u> — u(x) is given by u{x) = exp ( -e- 1 j f / ' [u£(z; x0)} dz^j . (2.26) Chapter 2. Metastability in Upward Propagating Flame 32 Cj = —6(f)jXL0U£ Upon integrating by parts, we obtain Lagrange's identity for any two smooth functions v and cf), {4>, L£v)w = {ecj>u)vx - e</>xuv) * + (Le<j>, v)w . (2.27) Next, we expand the solution v to (2.22) in terms of the eigenfunctions <f>j as 0 0 c-v = • (2.28) The coefficients Cj, obtained from (2.22) and (2.27) are -(RJ^ + x ' ^ d ^ A X - (2-29) As shown in §2.3, the severe indeterminacy in selecting the correct x0 for the equi-librium problem results in an exponentially small principal eigenvalue for (2.24). Since Le[dXou s] is uniformly small on [0,1] and dXou e is of one sign, this suggests that <j>0 is proportional to dXou s. Since Ao —> 0 as e —>• 0, a necessary condition for the solvability of (2.22) is that Co —•> 0 as e —> 0. Setting Co = 0 in (2.29) we obtain an asymptotic differential equation for .To = Xo(t): x'o (<f>o, dx,ue)w = (R, (j)0)w + ecf>0xuu£ |^  . (2.30) This differential equation will be valid for time intervals over which . T 0 0(e) and 1 — . T 0 >^ 0(e) (i.e., away from the endpoints). The metastable dynamics for (2.10) is then characterized by u(x, t) ~ u e[x; Xo(t)], where ue is defined in (2.20). The equilibrium value for . T 0 , corresponding to the equilibrium solution for u, is obtained by setting x'0 — 0 in (2.30), which yields the algebraic condition (R,<f>o)u = -eqb0xLoue |^  . (2.31) In §2.3 we will estimate </>0 and Ao as e —> 0, and in §2.4 we will evaluate the inner products in (2.30) and (2.31) asymptotically. These calculations-allow us to explicitly determine the equilibrium value for XQ from (2.31) and the form of the ODE for .T0(t) in (2.30). Chapter 2. Metastability in Upward Propagating Flame 33 2.3 Asympto t i c s and Numerics for the P r i n c i p a l Eigenpair We now estimate the principal eigenpair Ao, <f>o for ( 2 . 2 4 ) . Let fa be a trial function for 4>Q. Then, from Lagrange's identity ( 2 . 2 7 ) , we get Ao (<^ o,</>o)w = (Le<t>o,<i>a) + efax^fa | Q • ( 2 . 3 2 ) To a get a very rough estimate for Ao take <^ 0 = 1 for which L£l is exponentially small for 0{e) C I < 1 - 0(e). Then, substituting fa ~ dXouc into ( 2 . 3 2 ) , and using the fact that LO is exponentially small unless \x — xo\ — 0(e), it is readily clear that Ao = 0(e~cle) for some c > 0 . To get a precise estimate for Ao, we first replace ( 2 . 2 4 ) with the approximate equation Lefa — 0 . Then, in § 2 . 3 . 1 , we use boundary layer theory to construct fa for e —> 0 . The outer solution for fa is clearly fa ~ 1 (apart from a normalization constant). In § 2 . 3 . 2 we use ( 2 . 3 2 ) to estimate Ao, and in § 2 . 3 . 3 we compute Ao numerically. 2.3.1 Asympto t i c s for P r i n c i p a l Eigenfunction In the left boundary layer we let y = £ _ 1 x , fa(y) = fa(ey), and we expand fa as fa(y) = l + fao(y) + £fai(y) + ... . ( 2 . 3 3 ) Substituting ( 2 . 3 3 ) into Lefa = 0 , and using ( 2 . 1 2 ) , we obtain that 4>i0 satisfies <l>'io-[f(-xo + uio)(l + fa0)]' = 0, 0 < y < o o , ^ / 0(0) = - 1 , <f>,0(oo) = 0. ( 2 . 3 4 ) Comparing ( 2 . 3 4 ) with ( 2 . 1 3 ) , we conclude that fa0 — —dXoui0. Therefore, we have 4>'io(0) = -f'(-xo), faM~Mv-<)^ lV, as y-+oo . ( 2 . 3 5 ) Here and in the formulae to be derived below, the primes on the constants a;0, aro, vi and vr denote derivatives with respect to XQ. Chapter 2. Metastability in Upward Propagating Flame 34 Similarly, in the right boundary layer we let y = e _ 1 ( l - rc), <f>r(y) = (f>o(l - ey), and we expand cj>r as My) = i + My) + e</>T1(y) + • • • • (2-36) Substituting (2.36) and (2.17) into L£<f>0 = 0, we get to leading order that < + [ / H l - z o + W r o ) ( l + ^ r o ) ] ' = 0, 0 < y < oo, (2.37a) y r 0 (0) = - l , 0 r o(co) = O. (2.37b) Comparing (2.37) with (2.18), we have 0 r o = -dXouro . UR For similar reasons as outlined following (2.19) above, the expansions (2.33) and (2.36) hold only on the interval y = 0(eq), where — l / 2 < q < 0 . A composite expansion for <f>o, valid for x £ [0,1], is 0o(x) = 1 + K(.e~ lx) + <U£ _ 1 (1 - a)] + • • • . (2.38) This asymptotic eigenfunction can then be suitably normalized. In the derivation below to estimate Ao we require certain formulae involving the ratios (j)'ro/u'ro and (f>'lo/u'lo. The first identity is obtained by combining (2.13) and (2.34) to get Yy(^) = / " ( - * o + ti/,W/ 0 + l ) , 0 < y < c c , (2.39a) - f — = — - , -f— = vxy + o(l), a s y ^ o o . 2.39b) ui0(°) f(-xo) u'lo(y) ai0ut In a similar way, combining (2.18) and (2.37) we obtain d_ /# ^ » = - / " ( l - Z o + U r o H & o + l ) , 0 < y < o o , (2.40c ^o(O) = / ' (I - xo) ^ 0 (y ) _ , _ ( a ^ < (0 ) / ( 1 - x o ) ' < ( y ) K V arour v'ry-y 0  r> + 0 (1) , a s y ^ o o . (2.40b) Chapter 2. Metastability in Upward Propagating Flame 35 2.3.2 Asympto t i c s for the P r i n c i p a l Eigenvalue To estimate A 0 from (2.32) we choose the comparison function cf)0 = 1 and use (2.20) and (2.38) for u£ and <f>0, respectively. Substituting (2.38) and <j)0 = 1 into (2.32) we get Ao (1, </>0)u ~ (LA, l)u + (Lel, faX + (LA, KI ~ ^ o (0)w(l) - CI>'1o(0)LO(0) , (2.41) where (u,v)w = /Q1 uwudx and u> = u(x) is defined in (2.26). From (2.23a), we calculate LA = / " (u e ) (1 - fi«) . (2.42) To evaluate the three integrals on the right side of (2.41) we break the range of integration for each integral into the three regions x € [0,£p], x £ [e p , l — t p ] , and x € [1 — ep, 1]. Here the choice 1/2 < p < 1 gives an intermediate scaling between the outer and boundary layer regions and is needed to ensure that the leading order terms in the expansions for <f)0 and u6 in the boundary layer regions are asymptotically valid (see the remark following (2.19) above). To determine which integrals are asymptotically dominant we make the following observations: u> = 0(1) for \x — xQ\ = 0(e); OJ = t.s.t. for |a; — x0\ > 0(e); <t>i0 = t.s.t for x > 0(ep); 4>RO = t.s.t for 1 — x > 0(ep). From these considerations, we can reduce (2.41) asymptotically to Ao (1, l ) w ~ (h ~ o^(0V(0)) + (/2 - ^(0)^(1)) + h , (2.43) where h= I" UJ(\ + 4>h)LAdx, I2= f1 u(l + <j>ro)LAdx, I3= f1 " uLAdx. (2.44) We first estimate I\. Letting y = e~lx and using (2.20) and (2.42) we get h~ - uu'lo(y)f"[-x0 + ul0(y)} (1 + <t>M) dy • (2.45) J 0 Chapter 2. Metastability in Upward Propagating Flame 36 As shown in (A.5) of the Appendix, wujo is asymptotically constant on [ 0 , £ p _ 1 ] . Then, substituting (2.39a) into (2.45) we get r^"1 d h ~ -m>K(o) /' J 0 Therefore, using (2.39b), we obtain / i -^ o (0 )w(0)~-a ; (0 )u ; o (0 ) dy KJ 0 (2.46) (2.47) The integral I2 can be estimated in a similar way by using (2.40) and (A.7) of the Appendix to derive / 2 - ^ ( 0 ) W ( 1 ) ~ - « ( ! ) « ' (0) (2.48) Asymptotic formulae for w(0)wjo(0) and LO(1)U' (0) are given in (A.6) and (A.7) of the Appendix, respectively. Next we estimate 73. Using (2.42) and (2.20) we can decompose 73 as h ~ / 3 L + / 3 f l ! / 3 L = - / 1 _ £ P w / " (« e K, , 7 3 H = - ujf"{uc)urox dx . (2.49) JcP JsP For re £ [£p, 1—£p] we have from (A. 1) of the Appendix that u(x) = exp[—/(x — xo)/e](l + t.s.t.). Thus, using the decay behavior (2.13b) for ii{0 and ue ~ x — XQ, I3I becomes I3L ~ e _ 1a, 0i/ , / / " (x - a;o)e-' l'(a,)/e dx , fc,(x) = i/,x + / ( JeP X - XQ) , (2.50) where vx = - / ' ( - x 0 ) . Clearly fc/(x) has a minimum on [£p, 1 - £p] at x = ep. Thus, for £ -> 0, the dominant contribution to I3L arises from the region near x = ep. Let x = £p + s in 7 3 L and expand /i/(x) and /"(x - x 0) near x = £ p to get 7 3 L ~ e-1 aiovier ^~x^e 1 x exp / [ / " ( - x 0 ) + s n - i 0 ) + • • •] J 0 / " (-Xo ) ( 5 2 + 2£ P S + £2p)l ds. (2.51) Chapter 2. Metastability in Upward Propagating Flame 37 Then, using Laplace's method and the bound 1/2 < p < 1, we obtain as e -> 0 I: 3L ~ C%vle -S(-x0)ls ( ^ ) 1 / V ( -xo ) ] ^ - ^ n - ,o) + ^ /'"(-*<>) (2.52) A similar calculation, which we shall omit, can be used to calculate IZR as e —> 0. We find, hn ~ a ^ r e - ^ f [ ( ^ ) 1 / 2 [/"(I - xo)]1/2 - £ " - V " ( l - xo) - ^1*°)] • (2-53) Finally, we estimate (1,1)^ in (2.43). The dominant contribution to this integral arises from the region near x = x 0 . Assuming that /"(0) > 0, we obtain upon using (A. l ) of the Appendix and Laplace's method that /oo e-H*-x°V'dx~e1<260 (l-red1 + ---) , -oo #0 ' 27T ./"(0), 1/2 1^ /""(0) + 5[/'"(0)]2 8[/"(0)]2 24[/"(0)]3, (2.54a) (2.54b) To obtain our asymptotic estimate for A 0 we substitute (2.47), (2.48), (2.49), (2.52), (2.53) and (2.54) into (2.43). This leads to the following result: Propos i t ion 2.1 Let f(u) be smooth, convex, and satisfy /(0) = /'(0) = 0 with /"(0) > 0. Then, for e —> 0 and 0 < Xo < 1, the principal eigenvalue Ao for (2.24) satisfies [f"(o)}1/2 Ao 2e ai0vte -f(-xo)/e b,0 + e1/2bh + ••• bro + e1/2bri + •••]) (2.55) where K = [/"(-*o)] 1 / 2 . 2 . 1/2 <>» • (§) 6ro = [ / " ( 1 - X 0 ) ] 1 / 2 , 6r. = - ( f ) /'"(-so) (a , 0 n) ' ' 2 , 1/2 / " ( - x 0 ) a/0z/, / w ( l - x 0 ) ( a ^ ) ' / " (I - X 0 ) « r 0 ^ r (2.56a) (2.56b) Chapter 2. Metastability in Upward Propagating Flame 38 Here a/0, v\, aro, and vr, which depend on x0, are defined in (2.15) and (2.19). The •primes on these coefficients indicate differentiation with respect to XQ. This estimate for A0 characterizes the ill-conditioning of the equilibrium problem (2.11). Since A0 > 0 it also indicates that the equilibrium solution is marginally unstable. 2.3.3 Numerics for the Principal Eigenvalue We now verify (2.55) by comparing it with full numerical results for A0 computed from (2.24) for two choices of f(u) and for various values of XQ and e. Numerical methods to compute eigenvalues include the software packages S L E D G E and SLEIGN, and the N A G library code SL02FM (cf. [10], [11], [83], [84]). Our approach to compute A0 is to use the boundary value ODE solver COLSYS [6] with a suitable initial guess. To numerically evaluate the operator L£ in (2.24) we must first determine uE given in (2.20). In general, this requires us to numerically compute the boundary layer functions ui0 and uro satisfying (2.13) and (2.18) using COLSYS. Then, to compute A0 we re-write (2.24) as a first order system. Using A0 = 0 and (2.38) for </>0 as initial guesses, we found that COLSYS readily converged to the principal eigenvalue for (2.24). Example 2.1: Let f(u) = u2/2, which corresponds to the flame-front problem (2.5). For this problem, we calculate from (2.15) and (2.19) that a/0 = 2x0, v\ = x0, arQ = 2(1 — XQ), and vr = 1 — x0. Thus, (2.55) becomes AQ ~ e x0 /8e\^2' =-*o/2£ + (1 - so) (1 - xo) - p 8e\^2 -{l-x0f/2e (2.57) For this example, uto(y;x0) and uro(y;x0) can be found analytically as u M = xo - xotanh , uro(y) = (1 - x0) tanh (1 - x0)y - 1 (2.58) Chapter 2. Metastability in Upward Propagating Flame 39 Thus, the composite expansion u£, which is needed in (2.24), is obtained explicitly. Example 2.2: Here we choose the function f(u) = u - 2 + 4/(u + 2), which is not even. A careful calculation from (2.15) and (2.19) yields vi = -1 + vT — 1 Gf/0 = X 0 exp (2 - xoy ' 4 , aro = (1 - x 0) exp 4 ( 2 - x 0 ) - 2 log 4 ( 3 - x 0 ) - 2 l o g ( 2 5 - x 0 (2.59a) (2.59b) (3 - so)2 Substituting (2.59) into (2.55) gives the asymptotic result for A 0 . For this example the boundary layer functions, and hence L e , must be computed numerically. e A 0 (num.) A 0 (2.57) 1-term ratel A 0 (2.57) 2-term rate2 0.004 0 .2676xl0- n 0.3351xl0- n 0.252 0.2674xl0~ n -6.13x10" -4 0.006 0.5626xl0- 7 0.7465 x l O " 7 0.327 0.5619xl0- 7 -1.23x10" -3 0.008 0.7333xl0~ 5 0.1023xl0~4 0.395 0.7312xl0- 5 -2.88x10" -3 0.010 0.1276xl0- 3 0.1863xl0- 3 0.461 0.1269xl0- 3 -5.57x10" -3 0.012 0.8182xl0- 3 0.1247xl0- 2 0.524 0.8111X10"3 -8.70x10" -3 Table 2.1: Example 2.1: Comparison of asymptotic and numerical values for A 0 with f(u) = u2/2 and x 0 = 0.50. e Ao (num.) A 0 (2.57) 1-term ratel A 0 (2.57) 2-term rate2 0.002 0.2443 x l O " 1 1 0.3068X10"11 0.256 0.2442 x l O " 1 1 -3.52x10" -4 0.003 0.4168xl0- 7 0.5548 x l O " 7 0.331 0.4163xl0- 7 -1.31x10" -3 0.004 0.4893xl0- 5 0.6854xl0- 5 0.401 0.4878xl0~ 5 -3.08x10" -3 0.006 0.4920xl0- 3 0.7526 x l O " 3 0.530 0.4868 x l O " 3 -1.05x10" -2 0.008 0.4369 x l O " 2 0.7244xl0- 2 0.658 0.4290xl0- 2 -1.81x10" -2 0.010 0.1476X10-1 0.2680 x l O " 1 0.815 0.1458X10-1 -1.22x10" -2 Table 2.2: Example 2.1: Comparison of asymptotic and numerical values for A 0 with f(u) = u 2 /2 and x 0 = 0.35. In Table 2.1-2.4 we display the asymptotic and numerical results for A 0 for each of the two examples. In each of these tables, the second column gives the numerical results Chapter 2. Metastability in Upward Propagating Flame 40 e Ao (num. ) A 0 (2.55) 1-term ratel A 0 (2.55) 2-t erm rate2 0.00325 0.3888x10" - i i 0 .5267xl0- n 0.354 0.3791x10" •n -2.49x10" -2 0.00350 0.3210x10" -10 0.4405 x l O " 1 0 0.372 0.3124x10" 10 -2.68x10" -2 0.00400 0.9736x10 -9 0.1371xl0- 8 0.408 0.9448x10" -8 -2.96x10" -2 0.00450 0.1358x10 -7 0.1961xl0- 7 0.444 0.1314x10" -7 -3.24x10" -2 0.00500 0.1101x10 -6 0.1629xl0- 6 0.480 0.1063x10" -6 -3.45x10" -2 0.00550 0.6026x10 -6 0.9129xl0- 6 0.515 0.5798x10" -6 -3.78x10" -2 0.00600 0.2458x10 -5 0.3809xl0~5 0.549 0.2357x10" -5 -4.11x10" -2 Table 2.3: Example 2.2: Comparison of asymptotic and numerical values for A 0 with f(u) = u - 2 + 4/(u + 2) and x0 = 0.40 . £ A 0 (num.) A 0 (2.55) 1-term ratel A 0 (2.55) 2-term rate2 0.00225 0.4172xl0- 1 2 0.5519xl0- 1 2 0.323 0.4082 x l O " 1 2 -2.16x10" -2 0.00250 0 .9997xl0- n 0.1346xl0" 1 0 0.347 0.9769xl0- n -2.28x10" -2 0.00300 0.1137xl0- 8 0.1584xl0- 8 0.393 0.1107xl0- 8 -2.64x10" -2 0.00350 0.3239xl0- 7 0.4659xl0~ 7 0.439 0.3145xl0- 7 -2.90x10" -2 0.00425 0.1077xl0- 5 0.1621xl0" 5 0.506 0.1041xl0~ 5 -3.34x10" -2 0.00550 0.4114xl0- 4 0.6645xl0~ 4 0.615 0.3937xl0- 4 -4.30x10" -2 0.00650 0.2637xl0- 3 0.4489 x l O - 3 0.702 0.2500xl0- 3 -5.20x10" -2 Table 2.4: Example 2.2: Comparison of asymptotic and numerical values for A 0 with f(u) = u - 2 + 4/(u + 2) and x0 = 0.35 . for Ao, while the third and fifth columns show the asymptotic expansion (2.55) with one term and two terms in the pre-exponential factors, respectively. In the fourth and sixth columns we display the relative error rate = A°(^0 ~ Vnum.) _ Ao(num.) Here A0(asy.) denotes the asymptotic result with either one or two terms in the pre-exponential factors. • From these tables we observe that a two-term asymptotic expansion for the pre-exponential factor of AQ is certainly needed to obtain close quantitative agreement with Chapter 2. Metastability in Upward Propagating Flame 41 the numerical results for A 0 . A similar situation was found in [67] for some related problems with exponentially small eigenvalues. From Tables 2.1 and 2.2 we find that, in most cases, the relative errors for the two-term expansion in most cases are below 1 %, while they are only between 20 % and 70 % for the one-term expansion. Similar behavior is observed in Tables 2.3 and 2.4 for the relative errors for Example 2.2. 2.4 Der iva t ion of the Metastable Dynamics We now asymptotically evaluate the various terms in (2.30) to obtain an explicit ODE for xo = x0(t). The metastable dynamics for (2.10) is then given by u(x,t) ~ u£[x; Xo(t)], where uE is defined in (2.7). Similar considerations as given following (2.42) above show that (2.30) reduces asymp-totically to - x 0 ( l , l ) w ~ / 1 + / 2 + / 3 , (2.61) where the lj, for j = 1,2,3, are defined by h = f w(l + ^ 0 ) i 2 d x , I2 = C w(l + <f>ro)Rdx, h = C ' toRdx. (2.62) Here R is defined in (2.23b) and 1/2 < p < 1 gives the intermediate scaling used in §2.3. We note that the boundary term S<J)QXLOU£ in (2.30) can be neglected in comparison to each lj since it involves the product of the two exponentially small terms u and u£ at x = 0,1. To calculate R we substitute (2.20) into (2.23b) and use (2.13a) and (2.18a) to get R - [f(-x0 + «/o) - f'(ue)} ui0x + [f(l - x0 + uro) - /'(««)] urox . (2.63) The inner product (1,1)^ in (2.61) was calculated for e —• 0 in (2.54). We first estimate I-y. In the region 0 < x < ev we can approximate R by R ~ — /"(—Xo + ui0)xui0x. Substituting this expression into Ii and letting y = e~lx we obtain, Chapter 2. Metastability in Upward Propagating Flame 42 upon using (2.39a) and (A.5) of the Appendix, that J i ~ - £ / uu'i0yf"(-xo + «/„) (1 + <f,lo) dy J 0 -(oK(o)j^y^(J) * . (2.64) Then, since <j>'lo = - d ^ o and U'1Q < 0 we can re-write (2.64) after integrating by parts as h ~ eu(0)u'lo(0)dx £P-l reP- 1 y^gi-u'j o -jQ i o g ( - « ; o ) dy (2.65) — l oo as e Recall from (2.13b) that u\Q ~ —ai0vie~Uiy as y —> oo. Since £ p for 1/2 < p < 1 we can use this decay behavior to estimate log(—u\) at the upper endpoint y = e p _ 1 . In addition, we can add and subtract the term log (cn0vf) — v\y inside the integrand in (2.65) so that the resulting integral converges when the upper limit of integration is set to infinity. In this way, we obtain that _ ^ . . 2 p - 2 (2.66) Here v\ = dvi/dx0 = f"(-x0) from (2.15a), and = (3i(xQ) is defined by P' = ~dV0 0T N (~U'M) ~ l os( f l'o"z) + M  dy) • (2-67) A very similar calculation can be done to estimate I2 in (2.62). In the region 1 — ep < x < 1, we have R ~ -f"(l - x0 + uro)(x - l)urox in (2.62). Thus, upon using (A.7) of the Appendix and (2.40a), I2 becomes / 2 _ e r P - l uu'royf"(l -x0 + uro) (1 + <f)ro) dy rP-l d K0 ^< {0)Io  yd-y[t) dy-  (2-68) Then, since #.o = -dXou'rQ and u'TQ > 0 we can re-write (2.68) after integrating by parts as -eu(l)u'ro(0)dxo £ P - l reP- 1 y l o g « ) - / l o g K J dy J 0 (2.69) Chapter 2. Metastability in Upward Propagating Flame 43 Finally, using u'To ~ afQi/re UtV as y -> oo in (2.69) we obtain, in analogy with (2.66), that h ~ - e u , ( l K ( 0 ) (~i £2P~ 2 + • (2-70) Here v'r = -f"(l - x0) and 0r is defined by & = [log ( < ( y ) ) - log(a ro^,) + !/rj,] dy) . (2.71) Next we estimate I3 in (2.62). Using (2.63) we first decompose I3 as I3 ~ 7 3 i + J 3 f l , where /•l-e^ 7 3 L ~ w«i t a [/'(-x 0 + «/ 0) -f'(u')] dx, (2.72a) /•1-eP J 3 H ~ J uurox[f'(l-x0 + uro)-f'(ue)}dx. (2.72b) The dominant contribution to I3i arises from the region near x = ep. To calculate I3i we use (A. l ) of the Appendix to estimate OJ, (2.13b) to evaluate ui0x, and we expand / ' ( -xo + uio) - f'(ue) for x - • 0. This yields, hL ~ e" 1 ^!/ , fj" (xf"(-x0) + y / " ' ( - x 0 ) + • • C-fc«<*)/« dx , (2.73) where /i;(x) = vix + f(x — X o ) . It is clear that hi(x) is minimized on [ep, 1 — ep\ at x = ep. Therefore, we can use Laplace's method to evaluate (2.73) by expanding h\[x) as x —» 0. In this way, we obtain for e —> 0 that /3L ~ awe-H-**"' (l -  £^n-x0) + • • • ) . (2.74) A similar calculation, which we shall omit, can be done to calculate I3R as e —*• 0, with the result I3R ~ ~arovre-^-^ (\ - ^ - / " ( l - x0) + • • . (2.75) Then, J 3 in (2.62) is given by 73 ~ I3L + I3R. Chapter 2. Metastability in Upward Propagating Flame 44 Finally, an explicit O D E for x0 = xQ(t) is obtained by substituting (2.54), (2.66), (2.70), (2.74) and (2.75) into (2.61). The formulae (A.6) and (A.7) of the Appendix are used to evaluate u(0)u't (0) and u>(l)u' (0), respectively. This leads to our main result. Proposition 2.2 (Metastable Dynamics): Let f(u) be smooth, convex, and satisfy /(0) = /'(0) = 0 with /"(0) > 0. Then, for e —> 0 and t >> 1, the metastable dynamics for (2.10) is given by u(x,t) ~ u s[x; x0(t)], where u€ is given in (2.20) and Xo(t) satisfies the asymptotic nonlinear ODE x'0 ~ e" 1 /V (1 + ed1)~1 [aTour (1 + e0T) e^ 1 " * " ) / * - a,0v, (1 - ep,) e^'*^} . (2.76) The coefficients ai0, v,, arg, vr, Q\ and 0r, which all depend on XQ, are defined in (2.15), (2.19), (2.67) and (2.71). In addition, 60 and 9i are given in (2.54b). The following equilibrium result is obtained by setting x'0 — 0 in (2.76): Corollary 2.1 (Equilibrium): The (unstable) equilibrium solution to (2.10) of the form given in (2.7) is u ~ u E[x;x™] ; where x™ satisfies the nonlinear algebraic equation arour (1 + e$r) e-W-*°Ve ~ a,0u, (1 - ep,) e~ f^ s . (2.77) The special case f(u) = u2/2 corresponds to the flame-front problem (2.4) (or equiv-alently (2.5)). For this special case, a,0 = 2x 0, v, = xo, arQ = 2(1 — XQ) and vT = 1 — XQ. In addition, since u,0 and uro are given analytically in (2.58), we can calculate P, and Pr explicitly as * = 2 i ( f I 1 + * ) = • ft = - f i • ( 2 - 7 8 » In addition, when f(u) = u2/2, we have 6o = (2n)1^2 and 9\ = 0. Substituting these formulae into (2.76) yields the following explicit metastability result for (2.4) (or (2.5): Chapter 2. Metastability in Upward Propagating Flame 45 C o r o l l a r y 2.2 (Metastable F lame-Front Dynamics ) : For e —> 0 and t > 1, the tip x0 = x0(t) of the metastable parabolic-shaped flame-front interface for (2.4) satisfies the asymptotic nonlinear ODE (2.79) ; i - s o ) 2 + ^ j e - d - - ) ^ _ ( ^ + ^ ) e ^ / 2 £ Some trends can be observed from these results. Let XQ = Xo(0) denote the initial condition for (2.76). Then, since for x° < x™ (x° > x™) we have x'0 < 0 (x'0 > 0) from (2.76), it follows that x0(t) will not approach x^ as t —• oo, but instead will eventually hit the wall at x = 0 (x = 1). In addition, when 0(e) « i 0 « 1 - 0(e), (2.76) shows that x'Q is exponentially small and hence the motion is metastable. It is also clear that unless XQ is within an 0(e) neighborhood of only one of the exponentials on the right side of (2.76) is significant for e —* 0. Finally, in the case when f(u) is even, it is easy to see from the definitions of the coefficients in (2.77) that j3i — —(3r, ai0 = aTo and v\ = vr. Hence, in this case we have x™ — 1/2 as expected. 2.5 Compar i son of A s y m p t o t i c and Numer i ca l Results We now compare the asymptotic results (2.76), (2.77) and (2.79) with corresponding full numerical results computed directly from (2.5), (2.10) and (2.11) using the T M O L in §1.3. The metastability results (2.76) and (2.79) are valid only after the completion of an 0(1) transient period that describes the formation of the quasi-equilibrium solution (2.20) from initial data. As discussed in §2.1, a metastable quasi-equilibrium solution will not be formed for arbitrary initial data UQ(X). A sufficient condition on UQ(X) for metastability to occur is given in (2.6). To eliminate the effect of the initial transient, in the computations below we took u(x,0) — U£[X;.TQ] as the initial data for (2.10) and (2.5). Here u€ is the quasi-equilibrium profile given in (2.20) and x° £ (0,1) is the initial Chapter 2. Metastability in Upward Propagating Flame 46 zero of u. The value XQ is then used as the initial condition for the asymptotic ODE's (2.76) and (2.79) (i. e. xo(0) = .T°). With this initial condition, these ODE's are solved for t — t(x0) using a numerical quadrature. Asymptotic and numerical results for t = t(xo) are compared for f(u) = u2/2 and for the asymmetric f(u) of Example 2.2 in §2.3.3 given by /(«) = u - 2 + - i - . (2.80) For this latter form of f(u), explicit formulas for vi, a;0, ty, and aTQ are given in (2.59). However, the functions (3i(xo) and 0r(xo) in (2.76) are calculated from a numerical quadra-ture after first using COLSYS to solve for the boundary layer functions u'lo and u' . For the f(u) of (2.80) we have 60 = (2n)ll2 and 9X = 3/32 in (2.76). XQ t(num.) t(asy.) 0.3999125 0.3972259 0.3943236 0.3829263 0.3636165 0.3094434 0.2514594 0.2005907 0.1122723 0.196752251 x l O 5 0.554179455 x l O 6 0.997547019 x l O 6 0.197541881 x l O 7 0.235602159 x l O 7 0.245565007 x l O 7 0.245728124 x l O 7 0.245732411 x l O 7 0.245732918 x l O 7 0.192198642 x l O 5 0.541388563 x l O 6 0.974550456 x l O 6 0.193020273 x l O 7 0.230237584 x l O 7 0.240006441 x l O 7 0.240171229 x l O 7 0.240176186 x l O 7 0.240176776 x l O 7 Table 2.5: A comparison of the asymptotic and numerical results for the tip t = t(x0) of the flame-front for (2.5) with e = 0.004 and arg = 0.4. For f(u) = u2/2, in Table 2.5 and 2.6, we compare the asymptotic and numerical results for the tip t = t(x0) of the flame-front interface for e = 0.004 and e = 0.002, respectively. The initial tip location of the interface was x^ — 0.4 for e = 0.004 and a,'o = 0.3 for e = 0.002. The asymptotic and numerical results for the elapsed time agree to roughly within 2% for each of these examples. In Figure 2.7 we plot, at different times, Chapter 2. Metastability in Upward Propagating Flame 47 XQ t(num.) t(asy.) 0.2996765 0.2958542 0.2907243 0.2875337 0.2704028 0.2500379 0.2306506 0.1811032 0.0903085 0.110811891 x l O 7 0.109730472 x l O 8 0.180669282 x l O 8 0.205054498 x l O 8 0.245669157 x l O 8 0.250251570 x l O 8 0.250644804 x l O 8 0.250699164 x l O 8 0.250699778 x l O 8 0.108295000 x l O 7 0.107247855 x l O 8 0.176594370 x l O 8 0.200430705 x l O 8 0.240128133 x l O 8 0.244610251 x l O 8 0.244996311 x l O 8 0.245050448 x l O 8 0.245051150 x l O 8 Table 2.6: A comparison of the asymptotic and numerical results for the tip t = t(xo) of the flame-front for (2.5) with £ = 0.002 and xg = 0.3. the numerical solution to (2.5) for e = 0.002 with the initial data u(x,0) = ue(x; xg), where xg = 0.3. In Figure 2.8 we compare the asymptotic and numerical tip trajectories t = t(x0) for different initial conditions xg when e = 0.005. A logarithmic (base 10) scale is used for the vertical axis and the horizontal axis represents the parabolic tip location XQ. On this logarithmic scale, the asymptotic and numerical results are virtually indistinguishable. For the asymmetric f(u) of (2.80), in Tables 2.7 and 2.8 we give a similar comparison between the asymptotic and numerical results for t = t(x0) for e = 0.004 and e = 0.003, respectively. The initial zeroes xg of u are given in the captions of these Tables. For both values of e , the agreement between the asymptotic and numerical results is slightly closer than that for the case f(u) = u2/2. For the f(u) in (2.80), in Figure 2.9 we plot the numerical solution to (2.10) at different times when e = 0.004. In Figure 2.10 we compare some asymptotic and numerical trajectories for t = t(xo) for different initial conditions xg when e = 0.006. For the asymmetric /(it) of (2.80), we now verify the asymptotic result (2.77) for the equilibrium location x0 = x™ corresponding to the equilibrium solution u ~ ^(x'jX™). Chapter 2. Metastability in Upward Propagating Flame 48 x0 t(num.) t(asy.) 0.3999931 0.3999216 0.3983913 0.3904987 0.3750825 0.3202282 0.2574865 0.2020624 0.0898295 0.131068167 x l O 6 0.149606553 x l O 7 0.277759851 x l O 8 0.103529490 x l O 9 0.140377491 x l O 9 0.146999074 x l O 9 0.147019482 x l O 9 0.147019608 x l O 9 0.147019614 x l O 9 0.133057866 x l O 6 0.151644288 x l O 7 0.281609517 x l O 8 0.105057930 x l O 9 0.142599219 x l O 9 0.149374926 x l O 9 0.149396440 x l O 9 0.149396594 x l O 9 0.149396603 x l O 9 Table 2.7: A comparison of the asymptotic and numerical results for t = t(xo) for the asymmetric f(u) of (2.80) with e = 0.004 and x^ = 0.4. In Table 2.9 we compare asymptotic and numerical results for x™ at different values of e. The asymptotic result for x™ was computed from (2.77) using Newton's method. The numerical value for x™ was computed from (2.11) using COLSYS (cf. [6]). These full numerical equilibrium solutions are plotted versus x for various e in Figure 2.11 for the asymmetric f(u) of (2.80) as well as f(u) — u2/2. As expected, the asymptotic results provide a closer determination of the corresponding numerical result as e is decreased. Finally, we show how to recover the solution y(x,t) to (2.4) from u(x,t). Since u = — yx, we have y(x,t) = h(t) - r u(s,t)ds, (2.81) Jo where h(t) is to be found. To determine h(t), we substitute (2.81) into (2.4a) to derive h'(t) = -eux(0) + f1 [X u(s,t)dsdx . (2.82) Jo Jo Since (2.4a) is invai'iant under a constant shift in y, we can take h(0) = 0. Therefore, h(t) = ^-eux(Q) + £ j * u(s,t)dsdx^dt. (2.83) To determine h(t) during the metastable evolution we substitute u(x, i) ~ ue[x; xo(t)] into (2.81) and (2.83). Here ue is given by (2.7), where u; 0 and uTQ are given in (2.58). Chapter 2. Metastability in Upward Propagating Flame 49 x0 t(num.) t(asy.) 0.3499993 0.3499269 0.3493105 0.3476808 0.3380873 0.3005508 0.2503833 0.2012807 0.0884271 0.112330078 x l O 5 0.132980269 x l O 7 0.120121899 x l O 8 0.359540384 x l O 8 0.102874830 x l O 9 0.126006321 x l O 9 0.126203956 x l O 9 0.126205089 x l O 9 0.126205114 x l O 9 0.111349461 x l O 5 0.134879406 x l O 7 0.121708663 x l O 8 0.364335772 x l O 8 0.104310107 x l O 9 0.127873597 x l O 9 0.128078207 x l O 9 0.128079463 x l O 9 0.128079498 x l O 9 Table 2.8: A comparison of the asymptotic and numerical results for t = t(xo) for the asymmetric f(u) of (2.80) with e = 0.003 and xg = 0.35. This yields, y{x,t) = h ( t ) - ^ x 2 - x 0 x + 2elog2-2elog(l + e- X o x / £ ) +2e [log(l + e-V-x°Ue) - log(l + e-^o)(i-x)/e^ j ? (2.84a) where h(t) is defined by h(t) lo { " exoa-xo ) + £ ( 1 ° S 4 ~ 1 ) + \ x 2 ° - \ X 0 + 1 6 + 2 £ l 0 g ( 1 + e ' ( 1 _ S 0 ) / " + 2(1 -xo) 1 — tanh 2 1 - x 0 2e Jo I 6x 0 ( l — -2e2 1 g(l + e - ^ / £ ) g(l + e-(i-*o)/e> x 0 1 - x 0 dt + e(log4-l) + - 2 x 2 - l - x o + l\dt (2.84b) Here g(x) is the Dilogarithm function defined by g(x) = JQ j^jdt, which satisfies g(x) ~ 1 — x as x —• 1. To determine y(x,t) during the metastable evolution we solve the O D E (2.79) numerically and obtain h(t) from a numerical quadrature. Equation (2.84a) then yields y(x, i ) . In Figure 2.2 and Figure 2.12 we plot the metastable solution y(x,t) versus x with e = 0.0115 and e = 0.006, respectively, at several values of t. For the example in Chapter 2. Metastability in Upward Propagating Flame 50 1 > I I I 1 1 1 1 1 0.8 — 1 = 0 - L - - t=.250251570e8 t=.250696615e8 0.6 t=.250699782e8 0.4 "5" 0.2 0 s ' --0.2 \ --0.4 1 1 1 1 1 — 1 1 1 1 Figure 2.7: Plot of the numerical solution to (2.5) at different times. Here e = 0.002 and x°n = 0.3. Figure 2.2 it takes a time t w 117.1 for the tip of the parabola to move from its initial position = 0.45 to its final equilibrium state at x0 = 0. The height h(t) of the parabola increases by roughly 5.86 during this evolution. A similar observation was observed in the numerical computations of [77] (see Figure 3 of [77]). When e is decreased, the height h(i) can increase dramatically as shown in Figure 2.12. This can be explained from (2.84b) since for e -> 0 (2.85) Chapter 2. Metastability in Upward Propagating Flame 51 10 10 10 10 10" 10' 103 102 10' 10° • Numerical Asymptotic 0.1 0.2 0.3 0.4 0.5 X0 symmetric f(u) 0.6 0.7 0.8 0.9 Figure 2.8: Plots of asymptotic and numerical results for the tip location t = t(x0) of the parabolic-shaped flame-front interface for (2.5) with e = 0.005. A logarithmic (base 10) scale is used for the vertical axis. £ Xo (asy.) XQ71 (num.) 0.004 0.4412712 0.4412866 0.005 0.4420214 0.4420579 0.006 0.4427894 0.4428554 0.007 0.4435751 0.4436858 0.008 0.4443787 0.4445542 0.009 0.4452002 0.4454679 0.010 0.4460395 0.4464367 0.011 0.4468968 0.4474730 0.012 0.4477721 0.4485919 0.013 0.4486654 0.4498109 Table 2.9: The equilibrium location of x™ for the asymmetric f(u) of (2.80). Chapter 2. Metastability in Upward Propagating Flame 52 Figure 2.9: Plot of the numerical solution to (2.10) with the asymmetric f(u) of (2.80) at different times. Here e = 0.004, and = 0.4. Chapter 2. Metastability in Upward Propagating Flame 53 x0 Figure 2.10: Plots of asymptotic and numerical results for t = t(x0) for (2.10) with the asymmetric f(u) of (2.80) with e = 0.006. A logarithmic (base 10) scale is used for the vertical axis. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 09 1 v -0.41—- 1 1 > ' 1 1 1 1 1 I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 9 0 9 1 (a) (b) Figure 2.11: Plots of the equilibrium solutions to (2.11) versus x for various e: (a) For f(u) = u2/2, xg1 = | ; (b) For an asymmetric f(u) of (2.80), xg1 are given in Table 2.9. Chapter 2. Metastability in Upward Propagating Flame 54 0 0.1 0.2 0.3 0.4 0.5 O A 0.7 0.8 0.9 Figure 2.12: Plot of y(x,t) versus x given by (2.84) with e = .006 that approximates the metastable behavior in (2.4). (a) initial quasi-equilibrium solution y(x,t) with tip location x0 = 0.45 at t = 0 ; (6) quasi-equilibrium solution with xQ = 0.4 at t = 130745 ; (c) quasi-equilibrium solution with x0 = 0.3 at t = 136353; (d) final stable equilibrium solution at t > 136394. Chapter 3 Metastability in Slowly Varying Geometry Problems 3.1 Convection-Diffusion-Reaction Equations in Thin Domains In this chapter, we study two singularly perturbed evolution equations exhibiting meta-stable dynamics in a weakly inhomogeneous medium. The first problem we consider is the following generalized Ginzburg-Landau equation, which models the slow propagation of an internal layer in a thin channel ut = —{Aux)x + Q(u), 0<x<l, r > 0 , (3.1a) ux(0,t) = 1^(1,2) = 0, u(x, 0) = u0(x). (3.1b) Here e > 0 is a small parameter and A = A(x,s) > 0 is the local cross-sectional area of the channel, which is specified below. In addition, Q(u) is a smooth function with exactly three zeroes on the interval s+] located at u = s_ < 0, u = 0 and u = s+ > 0. Introducing the double-well potential V(u) by V(u) = — Q(n) drj, we assume that Q'(s±)<0, Q'(0)>0, V(s+) = 0. (3.2) A typical example is Q(u) = 2(u — u 3) for which s± — ± 1 and V[u) = |(1 — u 2) 2. The motivation for studying (1.1) is related to the problem of determining the con-ditions for the existence of stable spatially inhomogeneous steady-state solutions to the Ginzburg-Landau equation ut = Au + Q(u), x£D; dnu = 0, x € dD. (3.3) 55 Chapter 3. Metastability in Slowly Varying Geometry Problems 56 Here D is a bounded domain in RN and dn denotes the outward normal derivative to 3D. In a convex domain it is well-known that (3.3) does not admit a stable spatially inhomogeneous steady-state solution (cf. [28], [73]). However, this non-existence result does not hold for non-convex domains (cf. [45], [57], [73]). In Appendix B we show how (3.1) arises from an asymptotic reduction of (3.3) when D is a thin, axially symmetric domain as shown in Fig. 3.1. In this context, x represents the direction along the axis of the channel and A represents the local cross-sectional area of the channel. When A = 1, which yields a constant channel cross-section, it is well-known that the propagation of an internal layer for (3.1) is exponentially slow as e —> 0 (i. e. metastable) and that a stable spatially inhomogeneous solution for (3.1) does not exist (see [26], [39], [62], [109]). When A = 1, the metastability is a consequence of an exponentially small eigenvalue for the linearization of (3.1) around an internal layer solution. This exponential ill-conditioning suggests that the dynamics of an internal layer so-lution for (3.1) will depend very sensitively on the channel cross-section A, when A is slightly offset from the uniform value A = 1. In particular, exponentially small changes in A — 1 should influence the dynamics greatly. Therefore, in §3.2 we study (3.1) as e —>• 0 for an A(x\e) of the form A(x;e) = 1 +e»g(x)e-s~ld . (3.4) Here \i and d > 0 are constants and g(x) is smooth. If g"(x) < 0 then D is convex and we expect that (3.1) will have no stable spatially inhomogeneous equilibrium solutions. When g"(x) > 0 and 0 < d < dc, where dc is some constant, we show in §3.2 that (3.1) can have a stable spatially inhomogeneous equilibrium internal layer solution where the internal layer is located at a zero of g'(x). This phenomenon in which an internal layer or other localized structure is stabilized by a weakly inhomogeneous medium is called pinning. The effect of pinning of other localized structures such as vortices in Chapter 3. Metastability in Slowly Varying Geometry Problems 57 Figure 3.1: A cylinder of revolution with cross-section described in dimensional variables byR = RQF(X/L). superconductivity has been studied in [29], [70]. When g"(x) > 0 and d = dc, we show in §3.2 that the internal layer can be pinned at other locations in the interval [0,1]. In §3.2.1 we provide an asymptotic estimate for the principal eigenvalue Ao associated with the linearization of (3.8). In §3.2.2 we use the projection method to derive a differential equation for the location x0(t) of the internal layer, and we determine its limiting behavior as t —> oo. Finally, in §3.2.3 we compare our asymptotic results with corresponding results obtained from a full numerical solution of (3.1). The second problem we consider is the nonlinear convection-diffusion equation u t + [f{u)]x — c{x', e)h(u) — euxx , 0 < x < 1, t > 0 , (3.5a) u(0) = o_ , u(l) = o + ; u(x, 0) = UQ(X) . (3.5b) Chapter 3. Metastability in Slowly Varying Geometry Problems 58 Here 0 < £ < 1, a . > 0 and a+ < 0 are constants, and h(u) and f(u) are smooth. The flux function f(u) is assumed to be convex and satisfies / ( « + ) = / ( « _ ) , /(0) = / (0) = 0, uf'(u)>0 for (3.6) The function c(x; e) is chosen to be c(x;e) = -efig'{x)e-£~ld. (3.7) Here fi and d > 0 are constants and g(x) is smooth. A primary motivation for studying (3.5)-(3.7) is that for the special case when h(u) = u and f(u) = u2/2, this problem models transonic gas flow in a nozzle of cross-sectional area A(x;e) given by c(x;e) = -Ax(x;e)/A(x;e) (cf. [53], [51], [71], [94]). Hence, for t < 1, the cross-sectional area A(x;e) can be taken precisely as in (3.4). In this context, the nozzle is said to be divergent if g'(x) > 0 for all x, convergent if g (x) < 0 for all x, and convergent-divergent if g (x) has no definite sign. For Burgers equation (f(u) = u2/2) in a straight channel where g(x) = 0, it was shown in [61], [64] and [87] that there exists a unique and stable equilibrium shock layer solution centered at Xo = \ • It was also shown that for the corresponding time-dependent problem, a viscous shock, which gets formed from the initial data, tends toward the steady-state solution only over an asymptotically exponentially long time interval as e —> 0. This metastable behavior arises from the occurrence of an asymptotically exponentially small principal eigenvalue for the linearization of Burgers equation around the viscous shock solution. In view of this exponential ill-conditioning of Burgers equation, we expect that shock-layer solutions can be significantly altered by perturbing the differential operator by exponentially small terms. The effect of such spatially homogeneous perturbations were considered in [64]. Our primary goal in §3.3 is to study the pinning effect induced by the spatially inhomogeneous term c(x;e) in (3.5). In particular, we analyze the existence, stability Chapter 3. Metastability in Slowly Varying Geometry Problems 59 and dynamics of equilibrium and time-dependent shock-layer solutions to (3.5). In §3.3.1 we obtain an asymptotic estimate for the principal eigenvalue A 0 associated with the linearization of (3.5) around a shock-layer profile. In §3.3.2 we use the projection method of [110] to derive a differential equation for the location x0(t) of the shock-layer trajectory. We then analyze the equilibrium solutions of this differential equation and determine their stability properties. In §3.3.3 we illustrate the results for certain forms of g(x) when h(u) = u and f(u) = u2/2, modeling transonic nozzle flow, and we compare our asymptotic results with corresponding numerical results. Our results show that, under certain assumptions, there can exist stable steady-state shock-layer solutions along a convergent nozzle or in the convergent part of a convergent-divergent nozzle. In contrast, it was shown using a nonlinear stability analysis in [34] that when c(x; e) is independent of e and when the diffusive term euxx in (3.5) is absent, the corresponding inviscid problem does not admit stable shock waves in these nozzles. 3.2 A General ized Ginzburg-Landau Equat ion We now study (3.1) in the limit e -» 0 with A(x;e) as given in (3.4). A one-layer metastable pattern for (3.1) can be approximated by u(x,t) ~ uc e 1(x - x0(t)) where uc(z) is the heteroclinic orbit that connects s+ and s_, which satisfies u"( z) + Q(uc) = 0 , -oo < z < oo , uc(0) = 0 , as z •oo ; uc(z) ~ s+ — a+e — v+z as z —¥ oo , Here the positive constants z/± and a± are defined by "± = [-Q'(s±)]* , loga± = l o g ( ± 5 ± ) + / Jo ±u± 1 71 + l[2V(sp s-s± ds (3.8) (3.9a) (3.9b) (3.10) Chapter 3. Metastability in Slowly Varying Geometry Problems 60 We now look for a solution to (3.1) for t > 1 in the form u(x,t) = uc e~1(x — x0(t)) +w(x,t), (3-11) where w <C uc and wt << d t u c . The trajectory Xo = xo(t) gives the approximate location of the zero of u(x,t) during the metastable evolution. Substituting (3.11) into (3.1), and using (3.9), we obtain that w satisfies the quasi-steady problem Lew = A (-S^XQU'^Z) - e^wl) , 0 < x < 1, (3.12a) wx(0,t) = ux(0,t)-dxuc\x=0~-e-1a-u-e-e~1,'-xo , (3.12b) wx(l,t) = ux(l,t) - d x u c \ x = 1 ~ - e ^ a + u + e - ' ' 1 ^ 1 - ^ , (3.12c) where z — e~l[x — Xo(t)]. Here A is given in (3.4) and the operator LE is defined by Lew = e2 (Aiox)x + AQ\uc)w . (3.13) 3.2.1 The Eigenvalue Analys i s For a fixed x0 G (0,1), we now study the eigenvalue problem Lc(j) = \<f>, 0 < x < 1, (3.14a) ^(0) = <?U1)=0, (</>,<!>) = 1. (3.14b) Here (u,v) = uvdx . For this eigenproblem, the eigenvalues Xj for j > 0 are real and the principal eigenvalue Ao is exponentially small as e —* 0. To estimate Ao and the corresponding eigenfunction <j>o we use the trial function 4>o = w'c[e_1(x — x0)}. Then, upon integrating by parts, we derive Ao (<^ o, fa) = (Vo, ie^ o) - e2A<£o<?W|o • (3-15) Using (3.4) and (3.12) we estimate Lefa = e^2g\x)e-*~ldfax. (3.16) Chapter 3. Metastability in Slowly Varying Geometry Problems 61 Since L£(f>o is exponentially small and 0o is of one sign, we have that 0o ~ iVo0o away from 0(e) regions near the endpoints at x — 0 and x — 1, where N0 is a normalization constant. However, this approximate form for 0o does not satisfy the homogeneous boundary conditions in (3.14b) and so we cannot use it to calculate 0o(O) and 0o(l)-Instead, these quantities are calculated after constructing boundary layer profiles for 0o near each enclpoint. Since A is exponentially close to 1, the boundary layer analysis given in [109] for the case A = 1 can be used to calculate 0o(O) ~ 2N0a_u_e-€~1"-X0 , 0O(1) ~ 2N0a+v+e-£'1,/+^-x^ . (3.17) Then, since the dominant contribution to the inner product integrals arises from the region near x — xo, the left side of (3.15) is estimated as (0o, k) ~ iVo (0o, 0o) ~ e/30N0 ; (30 = f°° [u'c(z)}2 dz = f + [2V(u)}1/2 du . (3.18) J — CO J s_ Next, we use (3.16) to estimate ( 0 o , £ e 0 o ) ~ (4>o,Lefo) ~ N0e»+2e-*-ld u'c(z)g'(x0 + ez)v!'c(z) dz . (3.19) By using a Taylor series expansion for g (XQ + ez) we get CO (0o, Leio) ~ - W + V ' " 1 ' ' £ e"V"+1)(xo)7fc , (3-20) where the coefficients 7^  are defined by 1 z 0 0 The first two coefficients are readily calculated to be 7o = 0, 7i = / V 2 . (3.22) Moreover, if uc(z) is an even function then 72^  = 0. Finally, substituting (3.17)—(3.20) into (3.15), we obtain the following key estimate for AQ : A:=0 1 f°° 7* = -77 / < ( * ) < ( * ) * * <k, fc = 0 , l , . . . . (3.21N Chapter 3. Metastability in Slowly Varying Geometry Problems 62 Propos i t ion 3.1 (Exponent ia l ly Smal l Eigenvalue): Fore —• 0, the exponentially small eigenvalue of (3.14) satisfies A 0 = Ao(x0) ~ 2/? 0 - 1 {a 2 + ^e- 2 e _ 1 ^( 1 -^ ) + a i / / 3 . e - 2 £ " 1 ^ 0 } CO - t f e ^ e - - 1 " £ ekg^+lHx0)lk . (3.23) k=i Here v±, a± are defined in (3.10), (3Q is defined in (3.18), and 7*. is defined in (3.21). 3.2.2 The Metas tab i l ty Analys i s We now derive a differential equation for the location x0 = x0(t) of the internal layer trajectory. We first expand the solution w to (3.12) in terms of the eigenfunctions (pj of (3.14) as « ( x , * ) = f ; ^ i ( s ) . (3.24) The coefficients Cj, which are found by integrating by parts, are Cj = -e~x (AX'0U'c, <pj) - e (Axuc, - e2Awxcpj\1Q , j = 0 , 1 , . . . . (3.25) Since Ao —> 0 as e —->• 0, a necessary condition for the solvability of (3.12) is that Co —>• 0 as e —> 0. Setting c 0 = 0 in (3.25), we obtain the asymptotic differential equation for xQ = x0(t) e~lx0 (AUC, (po) ~ -e (Axu'c, (p0) - £2<p0Awx\l. (3.26) To obtain an explicit differential equation for Xo(t) we must evaluate the inner product integrals and the boundary terms in (3.26). The dominant contributions to the inner product integrals arise from the region near x = XQ. First, the boundary terms in (3.26) can be calculated asymptotically from (3.12) and (3.17) as e2A(p0wx\l ~ 2iY0£ ( - a X e - 2 < t - 1 " + ( i - * o ) + a2_vle-2e"v-x°) . (3.27) Chapter 3. Metastability in Slowly Varying Geometry Problems 63 Now to evaluate (A x u' c , fa) we use (3.4) and fa ~ Nou'c to get (Axu'c, fa) ~ Noe^e-'-1* / ° ° g'{x0 + ez)[u'c(z)]2 dz . (3.28) J — CO By using a Taylor series expansion of g(xo + ez) we obtain oo (Axu'c,fa) ~ N0e> 1+1e-e-ldY,ek9 {k+1)MPk, (3.29) fc=0 where the coefficients 0K are defined by 1 r00 0K = yJ_Ju's(z)}2zkdz , £ = 0 ,1 , . . . . (3.30) Upon integrating by parts, we can show that 0K = 27^+1 for k > 0, where 7 ^ is defined in (3.21). Next, for e —> 0, we estimate the left side of (3.26) to get £ _ 1 x 0 (Au'c, fa) ~ AO/SOXQ . (3.31) Finally, substituting (3.27), (3.29) and (3.31) into (3.26) we obtain our main result for the metastable dynamics associated with the generalized Ginzburg-Landau equation (3.1): Propos i t i on 3.2 (Metastable Dynamics ) : For e —> 0 and t ^> 1, a one-layer metastable pattern for (3.1) is represented by u(x,t) ~ uc [e~l(x — xo(t))], where the internal layer trajectory x0(t) satisfies the asymptotic differential equation x'0 ~ h(x0) = 260Q1 [alvle-2'-1"^1-*0) - a2_v2_e-2^1 00 -e^c-^Vo 1 £ e*/W*+1)(*o) • (3-32) //ere z/±, a± are defined in (3.10), 0K for k > 0 is defined in (3.30), and uc(z) is defined in (3.9). The following equilibrium result is obtained by setting x'Q — 0 in (3.32): Chapter 3. Metastability in Slowly Varying Geometry Problems 64 Coro l l a ry 3.1 (Equ i l ib r ium) : Fore —• 0 an equilibrium solution U(x;e) to (3.1) corresponding to a one-layer pattern is given by U(x;e) ~ uc [e~1(x — x™)], where uc(z) is defined in (3.9) and x1^ satisfies the nonlinear algebraic equation h(xo) = 0, i.e., alule-2''1^1-^ - alvle-2'-1"-** = ±e»+1 e^1 d JT, ek 8kg[k+l\x0) . (3.33) k=0 We now discuss the behavior of the equilibrium solutions for x0(t). We first observe that in (3.32), h(0) < 0 and h(l) > 0 for e —> 0. Thus, there exists at least one equilibrium value x™ for x0(t). The existence of any other equilibrium value for x0 depends on the constants d and u and the function g'(x). For example, when d > 0 is sufficiently large, the terms in (3.33) proportional to e~e~^d are insignificant and consequently, the equilibrium value for x0 is given uniquely by x0 ~ log a+v+ a-V-(3.34) Alternatively, when d > 0 is sufficiently small, the right side of (3.33) dominates the left side of (3.33) and, consequently, for e —> 0, (3.33) has a root near each zero of g (x). As shown in the examples below, when d is near some critical value so that the right and left sides of (3.33) balance as e —> 0, we can have equilibrium internal layer solutions centered at different points on the interval [0,1]. Although the only stable equilibrium solutions to (3.3) in a convex domain are con-stants, the generalized G-L equation (3.1) may admit stable spatially dependent equi-librium solution with an internal layer structure. Let x™ satisfy /i(xoTl)=0. Then, since 8k = 27fc+i , as seen by comparing (3.21) and (3.30), we can show that h'(x0n) = 2Ao(a,,0"), where Ao is given in (3.23). This shows that the decay rate for the differential equation (3.32) associated with infinitesimal perturbations about x™ is 2\™ , where A™ = Ao(x 0 n). This leads to the following criterion for the stability of the equilibrium internal layer so-lutions: Chapter 3. Metastability in Slowly Varying Geometry Problems 65 C o r o l l a r y 3.2 (Stabi l i ty of E q u i l i b r i u m ) : Let x™ satisfy h(x£) = 0 . Then the equilibrium solution to (3.1) has the form U ~ uc[e~1(x — x™)] and is stable (unstable) if A 0 (x 0 n ) < 0 (Xo(x^) > 0) . Here uc(z), A 0(x 0) and h(x0) are given in (3.9) , (3.23) and (3.32). Using this corollary, it follows that an equilibrium solution with an internal layer located at x 0 = x^ is unstable when g"(x™) < 0. Since g"(x) < 0 corresponds to a convex domain in higher dimensions, this result re-states the conclusion in [28] and [73] concerning the instability of non-constant steady-state solutions to (1.3) in convex domains. However, when g"(x™) > 0, then A^ can be negative for certain choices of p and d, resulting in a stable internal layer solution centered at xJJ7,. The key point to construct a stable equilibrium solution is to guarantee that (3.32) has multiple equilibria corresponding to simple zeroes of h(x0). Then, we must have exactly one stable equilib-rium of (3.32) between every two consecutive unstable equilibria. We will see from the examples below that this can be realized by selecting the cross-sectional profile A(x,e) (i. e. g(x)) appropriately. 3.2.3 Compar i son of A s y m p t o t i c and N u m e r i c a l Results We now compare the asymptotic results obtained above with the corresponding full numerical results computed directly from (3.1). We also show the existence of stable equilibrium solutions with an internal layer structure to the generalized G-L equation (3.1). In all of the calculations below, we have taken Q(u) = 2(u — u 3), for which a+ = a_ = 2, v+ = V- = 2 and us(z) = tanh(z). In addition, we calculate that f30 = 4/3, 7 i = 2/3, 73 = (TT 2 - 6)/36 and 7 2 f c = 0 for k > 0. Thus, (3.23) becomes Ao = A 0 (x 0 ) - 48 [e-^-Mi-^o) + e - 4 e - ^ 0 ] Chapter 3. Metastability in Slowly Varying Geometry Problems 66 g"(x0) + {4)(x0)e2 + ... (3.35) Noting that j3k = 2^k+1 , the differential equation (3.32) t> ecomes 0 h(x0) = 24e [e-^d-^o) _ e g'(x0) + 7T 2 - 6 24 9 {3)(x0)e2 + ••• (3.36) To check the validity of (3.36), we solved (3.1) numerically for a number of choices of g(x), three of which are described below. To compute numerical solutions to (3.1) we use a transverse method of lines approach approximation and then solving the resulting boundary value problems in space. More specifically, we convert the time-dependent problem (3.1) to a set of boundary value problems using the second order Backward Differential Formulas (BDF) [7], which we solve at each time step using the boundary value solver COLSYS [6]. Since the motion of the internal layer solutions is exponentially slow, we found it necessary to implement a time-stepping control strategy to efficiently track the solutions to (3.1) over long time intervals. To achieve this, we used the Z2-norm of the difference between the solutions of the second order and the third order B D F schemes as an error indicator to reject large inaccurate time steps or to enlarge unnecessary small time steps. See [100] for details of these algorithms, where they were used in a different context. The metastablity result (3.36) is valid only after the completion of an 0(1) transient period that describes the formation of an internal layer from initial data. In the compu-tations below we took u(x,0) = u c(e _ 1[x — XQ]) as the initial data for (3.1), where uc(z) is defined in (3.9) and x° £ (0,1) is the initial zero of u. To eliminate any unwanted transient effects we computed the solution to (3.1) with this initial data until t — 5. At this time, xg is reset to be the zero of u predicted by the numerical method. This (cf. [7]). This method is based on replacing the time derivative in (3.1) by a differ. •ence Chapter 3. Metastability in Slowly Varying Geometry Problems 67 new value for is used as the initial condition for (3.36). The differential equation (3.36) is then solved numerically for x0(t) using the initial value solver DP12 [27] and for t(xo) using a numerical quadrature, and the results are compared with corresponding numerical results for the zero of u computed from the finite difference scheme. x0 *(asy.) t(num.) 0.3991489 0.3902339 0.3459655 0.3013976 0.2646857 0.2024301 0.1683312 0.1350323 0.0753443 0.100998498xl05 0.111103064xl06 0.515111092xl06 0.818109359xl06 0.102010258xl07 0.127991981xl07 0.133126841 x l O 7 0.133755526 x l O 7 0.133805967xl07 0.100998673 x l O 5 0.111104540 x l O 6 0.515123233 x l O 6 0.818137253 x l O 6 0.102014660 x l O 7 0.127995306 x l O 7 0.133127901 x l O 7 0.133756144 x l O 7 0.133806497 x l O 7 Table 3.1: Example 3.1: A comparison of the asymptotic and numerical results for t = t(x0) for Q(u) = 2(u - u3) and g(x) = - | ( x - \ f with e = 0.05, fi = 0, d = 0.4 and x0(0) = 0.4 . E x a m p l e 3.1: Let g(x) = — | (x — | ) 2 , which corresponds to a convex domain. Then a solution to the equilibrium problem h(xo) = 0 for (3.36) is x™ = 1/2, independent of the constants /* and d. This is the only solution to h(xo) = 0, since g"(XQ) < 0 implies that h (x0) > 0 for x 0 € [0,1]. This unique equilibrium solution x™ = 1/2 is unstable since A 0 ( l /2) > 0 in (3.35). This conclusion is confirmed by the full numerical results shown in Table 3.1. This table displays the asymptotic and numerical results for the elapsed time as a function of the internal layer location Xo for e = 0.05, p = 0 and d — 0.4, when the initial location is xo(0) = 0.4. The results for t = t(xo) agree to at least four significant decimal places. E x a m p l e 3.2: We choose g(x) = | (x — | ) 2 , which corresponds to a non-convex domain. Again, = 1/2 is an equilibrium solution to (3.36) for any fi and d. However, Chapter 3. Metastability in Slowly Varying Geometry Problems 68 this solution can be stable depending on the values of /j, and d. From (3.35), a simple calculation gives A 0 ( l /2) = - | £ ^ + 2 e _ £ ~ l d + 96e- 2* - 1 . Let dc be the zero of A 0 ( l /2) as a function of d, i.e., dc = 2 — elog 192 -f- (ft + 2)elog£. Then, from Corollary 2.4, the equilibrium x™ = 1/2 is stable (unstable) when d < dc(d > dc). Given u = — 2 and £ = 0.1, we have dc « 1.4742. In Fig. 3.2 we plot the numerical solution to (3.1) at different times for d = 1.4 and d = 1.5. For d — 1.5 and xo(0) = 0.49 we observe that the internal layer located at xo(t) moves at an accelerating speed away from X™ = | and eventually, it collapses against the wall at x = 0 . Alternatively, for d — 1.4 and .To(0) = 0.45 the layer drifts toward its equilibrium location at x™ = | at an exceedingly slow rate. Thus, this example demonstrates the influence of the constants ji and d on the stability of the equilibrium solution. Comparisons between the asymptotic and numerical results for the internal layer trajectories are displayed in Table 3.2a for d = 1.4 and in Table 3.2b for d = 1.5. They agree to at least 3-4 significant digits. t x0(asy.) a?o(num.) 0.1009705 x l O 4 0.4500052 0.4500052 0.5824404 x l O 5 0.4503374 0.4503374 0.6115657 x l O 6 0.4541856 0.4541852 0.1266815 x l O 7 0.4600576 0.4600546 0.3232563 x l O 7 0.4790305 0.4790227 0.4510299 x l O 7 0.4875179 0.4875137 0.8933232 x l O 7 0.4981489 0.4981502 0.1586249 x l O 8 0.4999096 0.4999098 0.2243776 x l O 9 0.5 0.5 Table 3.2a: Example 3.2: A comparison of the asymptotic and numerical internal layer trajectories for Q(u) = 2(u - u3) and g(x) = \(x - \ f with e = 0.1, u = - 2 , d = 1.4 and x0(0) = 0.45 . By plotting h(x0) in (3.36) versus x0 we can show that x% = 1/2 is the only equilib-rium to (3.36) when it is unstable. Alternatively, if the equilibrium x$ = 1/2 is stable Chapter 3. Metastability in Slowly Varying Geometry Problems 69 x0 *(asy.) f(num.) 0.4899742 0.255661494xl05 0.255661329 x l O 5 0.4892876 0.679551367xl06 0.679517141 x l O 6 0.4758890 0.747594011 x l O 7 0.747469530 x l O 7 0.4519193 0.105770564x10s 0.105738718 x l O 8 0.4263499 0.113221195xl08 0.113185051 x l O 8 0.4015483 0.115273938x10s 0.115236708 x l O 8 0.3522899 0.116191551x10s 0.116153855 x l O 8 0.3018846 0.116312395xl08 0.116274632 x l O 8 0.2021744 0.116330380x10s 0.116292599 x l O 8 0.0794917 0.116330716x10s 0.116292938 x l O 8 Table 3.2b: Example 3.2: A comparison of the asymptotic and numerical internal layer trajectories for Q(u) = 2(u - u3) and g(x) = \(x - \ f with e = 0.1, fi = - 2 , d = 1.5 and ajo(0) = 0.49 . then (3.36) has two additional (unstable) equilibria that emerge from a pitchfork bifur-cation as d is decreased below d = dc. In particular, for the parameter values e = 0.1, fi = — 2 and d = 1.4 , we calculate that there are two other equilibria at x™ ~ 0.4435 and x^ K 0.5565. For a more general g(x), the set of equilibria to (3.36) consists, for e —> 0, of the zeroes of g'(x0) and probably one or two others near the endpoints provided that d < 2 min(^_x m , ^+(1 — XM)) , where xm and XM are the smallest and largest zeros of g'(x) on the interval [0,1]. Since the equilibrium solution closest to the endpoint x = 0 or x = 1 is unstable, a stable equilibrium a;™ must be near those zeros of g'(x) satisfying g'^x™) > 0. This analysis is illustrated in the next example. Example 3.3: We now consider g(x) = JQ(S — |)(s — |) ds , which has one maximum at x\ = | and one minimum at x2 = | . From the discussion before, since g"(x\) < 0 (g"(x2) > 0), we expect that when d > 0 is sufficiently small the equilibrium of (3.36) near X\ (x2) is unstable (stable). This is confirmed by the numerical results plotted in Fig. 3.3, where £ = 0.08, u = 0 and d — 0.2. Fig. 3.3a shows that the internal Chapter 3. Metastability in Slowly Varying Geometry Problems 70 (a) (b) Figure 3.2: E x a m p l e 3.2: Plots of the numerical solutions to (3.1) at different times with Q(u) = 2(u — it3) and g(x) = |(x - | ) 2 , where e — 0.1, fi = — 2 with initial condition u0(x) = u c (e _ 1 [x — x0,]) . (a) When d — 1.4 and x% = 0.45 the internal layer moves towards its equilibrium x™ — 1/2; (b) When d = 1.5 and XQ = 0.49 the internal layer moves towards the left and collides with x = 0. layer drifts slowly towards the stable equilibrium location at x™2 w 0.6608 when its initial location is at x0(0) = 0.4. However, in Fig. 3.3b, the internal layer with initial location xo(0) — 0.333 moves slowly towards the left and finally collapses against the wall at x = 0. This shows that there is an unstable equilibrium near x\ , which is calculated from (3.36) to be x™x w 0.3401. Corollary 2.3 and. 2.4 suggests that there is another unstable equilibrium x^ between x2 and the right endpoint. We compute from (3.36) that w 0.7762. To confirm this conjecture, we compute the solution to (3.1) numerically for two different initial locations of the internal layer and we plot the corresponding numerical results at different times in Fig. 3.4. From this figure we observe that when x0(0) > x ^ or x™2 < xo(0) < x™3 , the internal layer moves exponentially slowly away from x ^ until it eventually collides with the endpoint x = 1 or it reaches its stable equilibrium location at x ^ , respectively. In summary, this example has three equilibrium internal layer solutions. The ones located at x ^ and x ^ are unstable, and the other one located at x™, is stable. Chapter 3. Metastability in Slowly Varying Geometry Problems 71 (a) (b) Figure 3.3: Example 3.3: Plots of the numerical solutions to (3.1) at different times with Q(u) = 2(u - u3) and g(x) = f£(s - |)(s - §) ds, where e = 0.08, n = 0, d = 0.2 with initial condition Uo(x) = uc(e~l[x — XQ]). (a) When XQ = 0.4 the internal layer moves towards x^ « 0.6608; (b) When x^ = 0.333 the internal layer moves towards the left and collides with x = 0. For this example, in Table 3.3a and Table 3.3b we give a comparison between the asymptotic and numerical results for the evolution of the internal layers corresponding to Fig. 3.3a and Fig. 3.3b, respectively. In these tables the second column gives the numerical results for Xo(t) or t(x0) while the third and fourth columns show the corre-sponding asymptotic results from (3.36) with the one term and the two term expansions for the second pair of brackets in (3.36), respectively. Since it may happen that g'(xo) is close to zero during the evolution of an internal layer, the higher order term in (3.36) proportional to g"'(x0) can be quantitatively significant in some cases. In most cases, we find that the relative errors for the two-term expansion are below 0.002% in Table 3.3a and 0.02% in Table 3.3b, while they are only about 3% in Table 3.3a and 50% in Table 3.3b for the one-term expansion. Thus, a two-term asymptotic expansion for (3.36) is certainly needed to obtain close quantitative agreement with the numerical results. We finally remark that by taking g(x) to be a periodic function it is possible to con-struct a domain profile A(x,e) such that (3.1) has arbitrarily many (stable) equilibrium Chapter 3. Metastability in Slowly Varying Geometry Problems 72 t=0 - - t=.4141e4 - - t=.8919e4 •••• t>.7604e5 -t=0 I- - t=1549.8 | t=1567.0 t=1569.1 -0.2 --0.4 -0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b) Figure 3.4: E x a m p l e 3 . 3 : Plots of the numerical solutions to (3.1) at different times with Q(u) = 2(w - u3) and g(x) - J0 x(s - - §) ds, where e = 0.08, fi = 0, d = 0.2 with initial condition uQ(x) = u c(e _ 1[x — XQ]). (a) When x^ = 0.76 the internal layer moves towards x™2 « 0.6608; (b) When x\ — 0.79 the internal layer moves towards the right and collides with x = 1. solutions. 3 . 3 A Burgers-l ike Convection-DifFusion-Reaction Equat ion We now study (3.5) in the limit £ -» 0 with c(x;e) as given in (3.7). The viscous shock solution for (3.5) can be approximated by (x,t) ~ us \e  l(x - x0(t)) where the viscous shock profile u3(z) satisfies u's(z) = f[us(z)] ~ / ( « - ) , - o o < z < o o ; ua(0) = 0, us(z) ~ a_ — a_eu~z , z —> —oo; us(z) ~ a+ + a+e~u+z , z —> +oo The positive constants v± and a± are defined by 1 1 v± = T / ' ( a ± ) , log(T—) = T^± I" Ct± Jo ± lf(s)-f(<*±) v±{s-a±)\ (3.37) (3.38a) (3.38b) ds . (3.39) Chapter 3. Metastability in Slowly Varying Geometry Problems 73 t x0(num.) x0(asy.) 1-term x 0 (asy.) 2-term 0.4531527xl04 0.446231116 0.452053925 0.446234119 0.7487310 x l O 4 0.484045085 0.493797390 0.484052581 0.8965202xl04 0.503953307 0.515317905 0.503963248 0.1339888x10s 0.560669741 0.574424596 0.560683682 0.2881822xl05 0.650108735 0.658569607 0.650107041 0.4544450 x l O 5 0.660137084 0.666792280 0.660135732 0.5741542 x l O 5 0.660748955 0.667250946 0.660748242 0.1175766xl06 0.660847732 0.667319903 0.660847282 0.5677799 x l O 8 0.660847731 0.667319905 0.660847287 Table 3.3a: Example 3.3: A comparison of the asymptotic and numerical inter-nal layer trajectories for Q(u) = 2(u - u3) and g(x) = Jx(s - §)(s - §)ds , with e = 0.08, u = 0, d = 0.2 and xo(0) = 0.4. We now look for a solution to (3.5) in the form u(x, t) — i t . £ 1(x - X0(t)) + v(x,t) (3.40) where v <C us and vt <C dtua. The trajectory XQ — xo(t) gives the approximate location of the zero of u(x,t) during the metastable evolution. Substituting (3.40) into (3.5), and using (3.38), we obtain that v satisfies the quasi-steady problem evxx - [f'(us)v]x + ch'(us)v = -ch(us) - e~xxQu'a(z), 0 < x < 1 , (3.41a) v(0,t) = a_ - M ^ - e - 1 ^ ) ~ a_e-u-e~lx« , (3.41b) v(l,t) = a+ - us(e-l[l - x0]) ~ - a + e - " + e - 1 ( i - » o ) ? (3.41c) where z — £ - 1 ( x — x 0(t)). As in [87], it is convenient to transform (3.41) to self-adjoint form by introducing a new variable w(x,i) defined by >{x,t) = w{x,t)ip{z), xp(z) = K0) /<(0)]^ , z = e - 1 ( x - x 0 ) . (3.42) Chapter 3. Metastability in Slowly Varying Geometry Problems 74 x0 £(num.) £(asy.) 1-term t (asy.) 2-term 0.3327547 0.3316803 0.3058932 0.2457738 0.2015392 0.1500169 0.1145554 0.1014824 0.0747960 0.340885709 x l O 5 0.147718828 x l O 6 0.744277679 x l O 6 0.101699029 x l O 7 0.110200256 x l O 7 0.116129605 x l O 7 0.117455785 x l O 7 0.117521955 x l O 7 0.117546167 x l O 7 0.153239646 x l O 6 0.444448389 x l O 6 0.120475205 x l O 7 0.148559051 x l O 7 0.157152695 x l O 7 0.163121386 x l O 7 0.164451780 x l O 7 0.164518040 x l O 7 0.164542305 x l O 7 0.340916775 x l O 5 0.147738247 x l O 6 0.744387573 x l O 6 0.101710846xl07 0.110210654xl07 0.116139950xl07 0.117466741 x l O 7 0.117532979xl07 0.117557243 x l O 7 Table 3.3b: Example 3 . 3 : A comparison of the asymptotic and numerical inter-nal layer trajectories for Q(u) = 2(u - u3) and g(x) = J^(s - §)(s - |) ds , with £ = 0.08, fi = 0, d = 0.2 and xQ(0) = 0.333. Substituting (3.42) into (3.41), and using the asymptotic behavior of ij)(z) a s ^ ± c o , we find that w(x,t) satisfies Lew = e2wxx - V[x;e]w - -e^~x (ch(ua) + e'1 x'0u's(z)) , 0 < x < 1, (3.43a) w(0,t) ~ [a-f(a-)/v_]Se-°-ll'-*°'\ (3.43b) w(l,t) ~ - \ a + f { a + ) j v + } K - ^ ^ l - ^ l 2 . (3.43c) Here V(x;e) is defined by V(x; e)=l- [f(us(z))}2 + \f'[us(z))u's{z) - ech'[us{z)), (3.44) where z = £~1(x - x0) and c = c(x;e) is given in (3.7). 3 . 3 . 1 The Eigenvalue Analys i s For a fixed x0 £ (0,1), we now study the eigenvalue problem Le(j> = \<j), 0 < x < l , <t>(0) = </>(!) = 0, [fa4>) = \ . (3.45a) (3.45b) Chapter 3. Metastability in Slowly Varying Geometry Problems 75 Here (u,t>) = JQ 1 uvdx . For this eigenproblem, the eigenvalues Xj for j > 0 are real and the principal eigenvalue Ao is exponentially small as e —> 0. We now extend the analysis of [87] to give an estimate for Ao and for the corresponding eigenfunction 0o-We first define the trial function 0o by (j>o(x) = (z)u's(z), where z = £ _ 1(rc — XQ) and tb is defined in (3.42). Then applying Green's identity to 0o and 0o , and vising (3.7), we get Ao (0o, </>o) = (^o, L£(f>0) + e2<f>ox<f>o\l> (3.46a) where L£0o = -e^1g'(x)e-£'ldh'[us(z)]u's{z)/^(z). (3.46b) Since -Le0o is exponentially small and 0O is of one sign, we have that 0O ~ Ao0o, except near the endpoints at rc = 0 and x = 1. Here AVj is a normalization constant. We must modify 0o by inserting boundary layer profiles near the endpoints in order to satisfy the boundary conditions in (3.45b). These boundary layers can be analyzed in the same way as in [87] and from this analysis, we obtain that M O ) ~ -e-1N0v-[a-v_f{a_)]h-e~1,'-X0'2 , (3.47a) 0o,(l) ~ e-'Nov+la+v+fia+yh-^^1-^2 . (3.47b) Since the dominant contribution to each of the inner product integrals in (3.46a) arises from the region near rc = rc0 we can calculate (0o,0o) and ( 0 o , £ j 0 o ) using Laplace's method. As in [87], we estimate (0o, 0o) ~ N0 (0o, k) ~ eN0 Pju's{z)]2i>-2(z) dz = eA 0 (o_ - a+)/(a_) . (3.48) To calculate (0o,-L£0o) we use (0o,L£0o) ~ No (J>o, Le(j>0). Substituting 0o = V' - 1 ^! into this expression, and using (3.42) and u'3(0) — —/(a_), we derive (0o,X e0 o) ~ N0f(a_y+2e-s~ld f°° g'{x0 + ez)us(z)h'[us(z)} dz . (3.49) J—oo Chapter 3. Metastability in Slowly Varying Geometry Problems 76 A Taylor series expansion for g (XQ -f- ez) then yields CO (<f>o,Lefo) ~ -N0f{aAs^ 2e-^ dY,sklkg {k+1\xo), (3.50) fc=o where the coefficients jk in (3.50) are defined by 7* = - - / u's(z)h'[us{z)]zk dz , fc = 0 , l , . . . . (3.51) /C; J — oo Since we have assumed that h'(u) is bounded on the interval [a+,a_], the exponential decay of u's(z) as z —> ± 0 0 ensures that 7 ^ is finite for each k > 0. We can calculate 7 0 explicitly to get 7 0 = / I ( Q _ ) -h{a+). (3.52) Notice that if f(u) is even and /j(u) is odd we get 72^+1 = 0 for A; > 0. Finally, substituting (3.47), (3.48) and (3.50) into (3.46a) we obtain the following key estimate for A 0 : Proposition 3.3 (Exponentially Small Eigenvalue): For e —> 0, the exponentially small eigenvalue of (3.45) satisfies A 0 = Ao(x0) ~ _ 1 la+vl e-'-1*^-**) + a_v 2_e-E~ lv- x» a.- — a + 1 1 -e- ld CO - — £eW* + 1 ) (*o). (3-53) a- - «+ fc=0 Here v±, a± are defined in (3.39), and 7 ^ is defined in (3.51). 3.3.2 The Metastability Analysis We now derive a differential equation for the location XQ = Xo(t) of the internal layer trajectory. We first expand the solution iv to (3.43) in terms of the eigenfunctions <pj of (3.45) as 00 r(t) w(x,t) = J2^y<j>J(x). (3.54) i=o Ai Chapter 3. Metastability in Slowly Varying Geometry Problems 77 The coefficients r, , which are found by integrating by parts, are rj = ~x'o (<l>j, V , _ 1 w' s ) - e (</»j, ^ - 1 cfe) + e ^ ^ x l o , (3-55) where ?/> is defined in (3.42). Since Ao —> 0 as s —>• 0, a necessary condition for the solvability of (3.43) is that r0 —* 0 as e —• 0 . Setting r 0 = 0 in (3.55), we obtain the asymptotic differential equation for x 0 = x0(t) x'o (0o, 0 _ 1w' s) ~ - £ (0o, i>~lch)j + £2w<j)ox\l • (3.56) To obtain an explicit differential equation for x0(t) we must evaluate the inner product integrals and the boundary terms in (3.56). The dominant contributions to the inner product integrals arise from the region near x — Xo-First, we use (3.43b), (3.43c) and (3.47) to asymptotically calculate the last term on the right side of (3.56) as e2wcf>0x\l ~ iV 0 £/(o_) { a _ ^ _ e - £ _ 1 " - * ° - a+iz+e- 8 - 1^ 1-* 0*} . (3.57) Next, we evaluate the term on the left side of (3.56) as (0O, V * - 1 ^ ) ~ d V 0 ( o - - a+) / (a_) , (3.58) which is the same as (3.48). To evaluate the first term on the right side of (3.56), we use 0o ~ iV o 0 o and 0O = {z)v!s(z) to get e (0o ,V _ 1 c / i ) ~ N0e^1e'e~ldf(a_)j\l(x)h{us[e-1{x-x0)])dx. (3.59) Since h(us(z)) —> h(a±) exponentially as z —> ± o o , we can evaluate the integral in (3.59) by decomposing it as f1 g'{x)h(us[e-\x - x0)]) dx ~ h{a.) [g{x0) - g{0)} + h(a+) [g{l) - g(x0)} Jo rxo rl + / [h(ut) - h(a-)] g'(x)dx + / [h(us) - h(a+)] g'(x) dx . (3.60) Jo Jx0 Chapter 3. Metastability in Slowly Varying Geometry Problems 78 The integrands in the two integrals on the right side of (3.60) are localized near x = xo and can be evaluated using a Taylor expansion to get f1 g'Whiusle-^x - x0)]) dx ~ h{a_) \g(x0) - g(0)} + h{a+) [g{l) - g(x0)] Jo CO +eJ2ek0kg{k+i)(xo), (3.61a) k=0 where the coefficients 0k , for k = 0 , 1 , . . . , are defined by 0k = -g f (h[us{z)] - h{a-)) zk dz + ^ J" {h[ua{z)] - h{a+)) zk dz . (3.61b) Using integrating by parts it is readily seen that 0k = Jk+i , where 7^  was defined previously in (3.51). We also observe that when f(u) is even and h(u) is odd, then 02k = 0 for k > 0. Finally, substituting (3.57), (3.58), (3.61a) into (3.56) we obtain our main result for the metastable dynamics associated with (3.5): Propos i t ion 3.4 (Metastable Dynamics ) : For e —> 0 and t ^> I, the metastable viscous shock dynamics for (3.5) is represented by u(x,t) ~ us [£ _ 1 (x — x0(t))], where the internal layer trajectory x0(t) satisfies the asymptotic differential equation x'Q ~ M(x0) = f a - i / . e - ' " 1 " - * 0 - a+v+e-^1"^1-^} e*e-e~H x x L { a . ) [g{xo) - gm + h(a+) [g(l) - g(x0)] + e £ sk0kg{k+l)(xo)\. (3.62) I k=o ) Here the coefficients a± and u± are defined in (3.39), 0k for k > 0 is defined in (3.61b), and us(z) is defined in (3.38). The following equilibrium result is obtained by setting x'0 = 0 in (3.62): Coro l l a ry 3.3 (Equ i l ib r ium) : For e —>• 0 an equilibrium shock-layer solution to (3.5) is given asymptotically by U ~ us[e~1(x — x™)] ; where us(z) is defined in (3.38) and Chapter 3. Metastability in Slowly Varying Geometry Problems 79 .To = XQ™ satisfies the nonlinear algebraic equation M(xo) = 0, i.e., a_v_e-e~ lu-X0 -a+iz+e-' - 1 "^ 1 -* 0 ) = e " e _ e ~ l d x x \h(a-) [g(x0) - g(0)} + h(a+) \g(l) - g(x0)} + £ £ £k(3kg (k+i)(x0)\ . (3.63) To qualitatively understand the equilibrium problem for x'Q = M(x -o), we first note that M(0) > 0 and M(l) < 0 as e —> 0 when d > 0. Thus, there is at least one equilibrium solution for (3.62) as e —> 0. If d > 0 is sufficiently large, then the term in (3.62) proportional to e~s~ ld can be neglected and hence the unique root X™ of M(x0) — 0 is given asymptotically by ^ - l o g ( ^ ) . (3.64) Alternatively, if <i > 0 is sufficiently small then Af (xo) = 0 may have multiple roots for some choices of g(x). This will be illustrated below for some specific examples. Since 7 / ; = fik-i , it follows that M'(xo) = £ - 1Ao(xo) • Thus, the decay rate associated with infinitesimal perturbations about x™ ^ s £~ 1Xo( xon) • This leads to the following criterion for the stability of the equilibrium shock-layer solution. Corollary 3.4 (Stability of Equilibrium): Let x% satisfy M(x%) = 0. Then the equilibrium solution to (3.5) has the form U ~ us[s~1(x — x™)] and is stable (unstable) ifXo( xo) < 0 (V^o 1 ) > 0) • H e r e us(z), Ao(s0) and M(xQ) are given in (3.38) , (3.53) and (3.62). It is easy to show that a sufficient condition for M' (x 0 ) < 0 on xo 6 [0,1] as e —*• 0 is that g'(x0) [h(a_) - h(a+)} > 0, for all x0 £ [0,1] . (3.65) When this condition holds, the shock-layer solution centered at XQ = x™, where x™ is the unique root of M ( x 0 ) = 0, is stable for £ C 1. In particular, (3.65) is satisfied Chapter 3. Metastability in Slowly Varying Geometry Problems 80 when h'(u) > 0 for u £ [OJ+,O:_] and g (x) > 0 on x € [0,1], which models a diverging nozzle flow as discussed following (3.7) above. More generally, a sufficient condition for the stability of a root x™ of M(x0) = 0 is that (3.65) holds at x0 — x™. This stability conclusion for an internal layer solution is consistent with [53]. However, when g'(x0)[h(a^) — h(a+)] < 0 on [0,1], the stability of an equilibrium internal layer solution can only be determined by explicitly calculating the sign of Xo(x0 n), where x™ is a root of M(x0) = 0. In this case, multiple equilibrium solutions for x'0 = M(x0) are possible (see the examples below). We now illustrate the existence of multiple equilibria in this case when d > 0 is sufficiently small and e —> 0. Let's suppose that g'(xo) < 0 on [0,1] and /i(a_) > h(a+) > 0. Then, when d > 0 is sufficiently small and e —> 0, there is a root x ^ of M(x0) = 0 that is O(e) close to the unique solution of /i(a_) [g(x0) - g(Q)] + h ( a + ) [g(l) - g(x0)} = 0 . (3.66) The assumption ^'(x0n2)[/i(o'_) — h(a+)] < 0 then yields that M'(x^) > 0 when d > 0 is sufficiently small and e —>• 0. Hence, this root is unstable. However, when d > 0 and e -» 0, we calculate that M(0) > 0, M(l) < 0, Af'(0) < 0, Af ' ( l ) < 0. Hence, there must exist additional roots x^ and x^ to M(x0) = 0 that satisfy 0 < x ^ < x^ and O^T < ^ol < 1 for w h i c n ^ '(^oi) < 0 a n d  M'i x03) < °- T l l u s 5 t h e s e additional roots are stable equilibria of x'Q = M(XQ) and the profiles us [£ _ 1 (x — x^)] and us [e _ 1(x — x^)] correspond to stable internal layer solutions. Notice, as d —•> 0 + , xg^ —> 0 and xgg —»• 1 so that these internal layer solutions become stable boundary layer solutions in agreement with the analysis in [53] for the case d = 0. 3.3.3 Compar i son of A s y m p t o t i c and Numer i ca l Results We now compare the asymptotic results obtained above with the corresponding full numerical results computed directly from (3.5). For all of the calculations below we Chapter 3. Metastability in Slowly Varying Geometry Problems 81 consider the case f(u) = u 2/2, h(u) = u, —a+ = Q _ = a. (3.67) As discussed in [94], and following (3.7) above, this problem models the transonic flow through a nozzle of cross-sectional area A(x;e) = 1 + e^e~ s ldg(x). We will consider nozzles of different cross-sectional areas by varying the term g(x) in (3.53), (3.62) and (3.63). Recall that the nozzle is said to be divergent if g'(x) > 0 for all x, convergent if g (x) < 0 for all x, and convergent-divergent if g (x) has no definite sign. We use a similar numerical method as described in §3.2.3 above to compute numerical solutions to (3.5) and to compare with the corresponding asymptotic results. When f(u) = u 2/2 and « _ = — a+ — a, we have a± = 2a u± = a and us(z) = —a tanh(o:z/2). We also calculate that ( 7 0 , 7 1 , 7 2 , 7 3 , . . . ) = (2o, 0,7r 2/3a, 0 , . . . ) . From (3.53) the principal eigenvalue XQ(XQ) satisfies In addition, since (5k = jk+1 for k > 0, (3.62) becomes X'Q ~ M(x0) = a (e-£~laxo - e - e _ 1 « ( i - ^ o ) j 7 T 2 £ 2 (3.68) - e " e - e l d (3.69) 2 J 1 602 In most cases, the higher order terms in the square brackets on the right sides of (3.68) and (3.69) make only very minor improvements to the results. Thus, except when specified otherwise, they are ignored when making the comparisons below. Example 3.4: We first consider a divergent nozzle, where g(x) = Cx for some C > 0. From (3.68) and (3.69) the only equilibrium value for x0(t) is x™ = \ and the principal eigenvalue A 0 ( l /2) is always negative for any /.i and d > 0, C > 0. Thus, from Corollary 3.3 and 3.4, there is a unique shock-layer solution centered at x = \ and it Chapter 3. Metastability in Slowly Varying Geometry Problems 82 is stable. This agrees with the conclusion in [34] and [71] that flows along a divergent nozzle are always stable. The case C = 0 was well-studied in [61], [64] and [87] and comparisons between asymptotic and numerical results can be found in [64] and [87]. E x a m p l e 3.5: We now consider a convergent nozzle, where g(x) = Cx for some C < 0. In this case, x™ = 1/2 is still a root of M(x0) = 0 in (3.69), but now from (3.68) we calculate A 0 (l/2) as A 0(l/2) ~ - 2 a W £ ~ l a / 2 - e'e-'^C . (3.70) Thus, the stability of x 1^ = 1/2 is determined by the values of ji, d and C . For example, if fi = — 1 and d = a/2, then the shock layer located at x™ = | is stable (unstable) if C > -2a 2 (C < -2a2). If u = -1 and C = -2a 2, then it is stable (unstable) if d > a/2 (d < a/2). These stability results are fully confirmed by the numerical results displayed in Table 4 and Table 5. In these tables, we give the asymptotic and numerical results for xo(t) in the second and third columns respectively, and the error representing the difference between the asymptotic and numerical results in the fourth column. The asymptotic results agree with the numerical ones to at least five decimal places. From these tables we observe that when the equilibrium x 1^ = | is unstable, the shock layer will move away from x™ = | to somewhere else, but not to the endpoints x = 0,1. This suggests the existence of other stable equilibria for (3.69). For this example, it is easy to show that when = | is unstable (i.e., Arj(|) > 0), then M{XQ) has exactly two more zeros that are symmetric about x™ = | . They correspond to two stable equilibrium values for xo(t). When x 1^ — | is stable, then it is the only zero of M(XQ) . This analysis is illustrated by plotting M(x0) in Fig. 3.5. Therefore, there exists either one or two stable steady-state solutions of the form u ~ us[e~ l(x — x 1^)] along the convergent nozzle we are considering. The analysis of [34] Chapter 3. Metastability in Slowly Varying Geometry Problems 83 t x0(t)(asy.) x 0 ( i ) (num.) error .7466667 x l O 1 .212682157 .212676330 .582x10" -5 .5013333 x l O 2 .236113084 .236101298 .117x10" -4 .3754166xl0 3 .285065717 .285059434 .628x10" -5 .3879890 x l O 4 .352169342 .352167776 .156x10" -5 .2523277xl0 5 .404404274 .404403590 .683x10" 6 .4144176xl0 6 .465271835 .465271222 .613x10" 6 .3061509xl0 7 .489469516 .489468465 .105x10" 5 .8745862 x l O 7 .496984897 .496983905 .991x10" 6 .8407137x10 s .499999999 .499999999 -.204x10" -9 Table 3.4a: E x a m p l e 3.5: A comparison of the asymptotic and numerical shock layer trajectories for (3.5) with f(u) = u 2 / 2 and g'(x) = —1.9. Here e = 0.03, ft = —1, d = 0.5, a = 1 and x°0 = 0.205712482. for the inviscid problem ut + uux — c(x)u = 0 , (3-71) proved that standing shock waves in a convergent nozzle (i.e. c(x) > 0 for all x) or in the convergent portion of a convergent-divergent nozzle are unstable. Our example has shown that the effect of viscosity and the boundary conditions in (3.5) allows for the existence of a stable standing wave in a convergent nozzle when c(x) is replaced by the form in (3.7). E x a m p l e 3.6: Next, we let a — 1 and consider a convergent-divergent nozzle where g(x) = (x — a) 2 , and the constant a satisfies | < a < 1. Now the algebraic equation 2g(xo) — g(0) — g(l) = 0 has one root x* £ (0 ,a) . If we choose d so that 0 < d < x* , then it is easy to see that for e —> 0 the function M(XQ) in (3.69) wi l l have three zeros XQ\ , x^ and x g 3 , satisfying 0 < < x ^ < x* and a < x ^ < 1. These zeros are illustrated in F ig . 3.6 for the parameter values e — 0.04, a = 1, /.i = — 1, d — 0.171244968 and a — 0.8. From the discussion following Corollary 3.4, it is clear that x ^ and x ^ are stable equilibria and that x ^ is unstable. In F ig . 3.7, we verify Chapter 3. Metastability in Slowly Varying Geometry Problems 84 t x0(£)(asy.) X o ( i ) ( n u m . ) error .2988174xl04 .494997385 .494997382 .346x10" -8 .1725713xl06 .494847418 .494847363 .548x10" -7 .4950768 x l O 7 .489521616 .489522250 -.633x10 -6 .7862987xl07 .486497884 .486497678 .206x10" -6 .9804466 x l O 7 .485206361 .485204630 .173x10" -5 .1465816xl08 .483947388 .483944120 .326x10" -5 .1659964x10s .483811685 .483908521 .316x10" -5 .2242408 x10 s .483701958 .483699155 .280x10" -5 .8697827x10s .483689263 .483686417 .284x10" 5 Table 3.4b: E x a m p l e 3.5: A comparison of the asymptotic and numerical shock layer trajectories for (3.5) with f(u) = u 2/2 and g'(x) = —2.1. Here £ = 0.03, fi = —1, d = 0.5, a = 1 and x° = 0.494999996. the stability of x ^ and x ^ by plotting the numerical solution to (3.5) at different times for two initial values x°. The two initial values x° in Fig. 3.7a and Fig. 3.7b are so close that there is only one unstable equilibrium for xo(t) between them, which is x ^ . E x a m p l e 3.7: Finally, we give an example to illustrate that it is possible to con-struct a nozzle geometry to guarantee an arbi trary number of steady state internal layer solutions. We take g(x) = s in(n7rx) where re is a positive integer. Let x* = ijn for i = 0, . . ,n be the i-th zero of g(x). In this case, the differential equation (3.62) for xo(t) becomes x n ~ a -e  1ax0 _ -e  la{\-x0) ( s in(n7rx 0 ) , ( = 1 — rcW 6 a 2 + (3.72) It is easy to see if d is chosen such that 0 < d < X j , then for e —• 0, there exists N = n + (n — l)mod2 equilibria x^ , i = 1 , 2 , . . . , ^ . Among these equilibria, x^ for i = 1, 3, . . . , iV are stable and the rest are unstable. Note that if d > x^, then the number of the equilibria may be less than N . We now implement a numerical experiment to illustrate this analysis. We choose n = 4, a = 1, /_i = 0, d — 0.19 and £ = 0.02. In this case, we have three stable Chapter 3. Metastability in Slowly Varying Geometry Problems 85 (a) (b) Figure 3.5: E x a m p l e 3.5: Plots of M1{x0) = a(-e~ £ W 1 - ^ ) + e - « 1 ( s o i 1 Q i m e s ) and A^xr j ) = e^e - 5 rfC(x0 — |) (dash lines) versus x 0 , where a = 1, e = 0.03, = — 1 and C = —2. The intersection(s) of M i ( x 0 ) and M 2 ( x 0 ) is the zero(s) of M ( x 0 ) . (a) d = 0.55: we have A 0 ( | ) < 0, and the only equilibrium x™ = | is stable; (b) d = 0.45: we have three zeros of M ( x 0 ) given by x ^ « 0.3902, x ^ = 0.5 and xg^ « 0.6097. Here X 0 2 ~ O-^  is unstable and the rest are stable, equilibria at x^ « 0.2044, x ^ = 0.5 and x ^ as 0.7955, and two unstable equilibria at xgJj « 0.2440 and x ^ as 0.7559, which we compute from (3.72) using a two-term expansion for (. We choose the initial values xg — 0.24 and xg = 0.25 when computing the numerical solution to (3.5). The asymptotic and numerical results are shown in Tables 6a and 6b. These tables also display the asymptotic result (3.72) with both the one-term and the two-term expansions for ( . The error terms in the fourth and sixth columns represent the difference between the asymptotic and numerical results. Since the higher order terms in ( in (3.72) are significant, we observe from Table 6 that a two-term expansion for n is certainly needed to obtain close quantitative agreement with the numerical results for xQ(t). Finally, in Fig. 3.8 we plot the shock layer evolution corresponding to the data in Table 6, which shows that XQ\ and * ^ 0 3 ^ ^ * ^ stable equilibria, while XQ2 is an unstable equilibrium. Chapter 3. Metastability in Slowly Varying Geometry Problems 86 t x0(i)(asy.) x0(i)(num.) error .1270000xl02 .216714648 .216706869 .777x10--5 .4765000 x l O 2 .235153498 .235142419 .110x10--4 .3321062xl03 .281796238 .281790749 .548x10" -5 .1271548xl04 .320208345 .320206583 .176x10" -5 .1494813x10s .392954938 .392955455 -.517x10--6 .1429582xl06 .457781790 .457782599 -.808x10" -6 .5020859xl06 .488305699 .488306411 -.712x10" -6 .1608928xl07 .499637394 .499637666 -.271x10" -6 .5455925 x10 s .500000000 .499999999 .582x10- 9 Table 3.5a: Example 3.5: A comparison of the asymptotic and numerical shock layer trajectories for (3.5) with / ( « ) = u2/2 and g'(x) = - 2 . Here e = 0.03, \i = —1, d = 0.55, a = 1 and = 0.205646164. t x0(t)(asy.) x0(i)(num.) error .3895482 x l O 3 .494967222 .494967221 .946x10" -9 .7976059 x l O 4 .494295191 .494295125 .658x10--7 .3291548x10s .491389800 .491389298 .502x10" -6 .8279433 x10 s .480483494 .480480756 .273x10--5 .1491332xl06 .445338979 .445330373 .860x10--5 .1690847xl06 .429122276 .429116007 .626x10" -5 .2249490 x l O 6 .395546688 .395541088 .560x10" -5 .3072491 x l O 6 .390353158 .390348796 .436x10" -5 .2905679 x l O 1 0 .390284916 .390281195 .372x10--5 Table 3.5b: Example 3.5: A comparison of the asymptotic and numerical shock layer trajectories for (3.5) with / ( « ) = u2/2 and g'(x) = -2. Here z = 0.03, ft = - 1 , d = 0.45, a = 1 and xg = 0.494999513 . Chapter 3. Metastability in Slowly Varying Geometry Problems 87 Figure 3.6: Example 3.6: Plots of M^XQ) = a(-e~s W i - * o ) + g - * - 1 ^ ) ( s o l i d l i n e s ) and M 2 ( x 0 ) = e^e~^d{{xQ - a)2 - \[a2 + (1 - a)2] + 7r 2Q- 2e 2/3) versus x0, where £ = 0.04, a = 0.8, a = 1, ^ = -1 and d = 0.1712 . . . . The intersections of M^XQ) and M2(x0), which are the zeros of M(x0), are reft ~ 0.1286 , x$2 « 0.2077 , and sg| « 0.9122 . Here the equilibrium x£2 is unstable, and the other two are stable. Chapter 3. Metastability in Slowly Varying Geometry Problems 88 Figure 3.7: E x a m p l e 3 . 6 : Plots of the numerical solutions to (3.5) at different times with f(u) = u 2/2 and g(x) = (x - a) 2 , where e = 0.04, a = 0.8, a = 1, fi = - 1 , d = 0.1712... and initial condition u0(x) = us[e-\x - x®)}. (a) x° = 0.2: shock layer-moves towards left; (b) a:° = 0.25: shock layer moves towards right. t x0(t)(num.) (3.72) 1-term error1 (3.72) 2-term error2 .1133545xl03 .239718703 .239610666 -.108x10--3 .239726721 .801x10" -5 .5512484xl03 .238497828 .237864222 -.633x10" -3 .238543918 .460x10" -4 .1135106xl04 .236387221 .234724657 -.166x10" -2 .236505117 .117x10" -3 .2580156xl04 .228182044 .222194243 -.598x10' -2 .228599741 .417x10" -3 .4255100 x l O 4 .214978265 .206459679 -.851x10" -2 .215700702 .722x10" -3 .6061413xl04 .206403299 .201517420 -.488x10" -2 .206902782 .499x10" -3 .8360356 x l O 4 .204368747 .200989316 -.337x10" -2 .204672929 .304x10" -3 .1606592xl05 .204187662 .200966538 -.322x10" -2 .204461290 .273x10" -3 .6239090 x l O 5 .204187642 .200966538 -.322x10" -2 .204461248 .273x10" -3 Table 3.6a: E x a m p l e 3 . 7 : A comparison of the asymptotic and numerical shock layer trajectories for (3.5) with f(u) = u 2/2 and g{x) = sin(47rx). Here e = 0.02, a = 1, H = 0,d = 0.19 and x°0 = 0.239991525 . Chapter 3. Metastability in Slowly Varying Geometry Problems 89 t rc0(i)(num.) (3.72) 1-term error1 (3.72) 2-term error2 .9389256xl02 .250384983 .250386816 .183x10" - 5 .250384917 -.664x10" - 7 .5804413 x l O 3 .252677437 .252755503 .780x10" - 4 .252672223 -.521x10" - 5 .1650848xl04 .261513735 .262616867 .110x10" -2 .261438939 -.748x10" - 4 .3207804xl04 .296383726 .306534177 .101x10" -1 .295721003 -.662x10" - 3 .4969111 x l O 4 .398142523 .424482167 .263x10" -1 .396102752 -.204x10" - 2 .6545529 x l O 4 .469337132 .481526941 .121x10" -1 .468203669 -.113x10" -2 .9610786xl04 .497704786 .498961715 .125x10" -2 .497567194 -.137x10" - 3 .1570298xl05 .499987099 .499996630 .953x10" -5 .499985677 -.142x10" -5 .6945472 x l O 6 .5 .5 .573x10" -9 .5 -.274x10" - 9 Table 3.6b: E x a m p l e 3.7: A comparison of the asymptotic and numerical shock layer trajectories for (3.5) with f(u) = u 2/2 and g(x) = sin(47ra:). Here e = 0.02, a = 1, fi = 0,d = 0.19 and x°0 = 0.250022546. Chapter 4 Phase Separation Model s in One Spatial Dimens ion 4.1 The Viscous C a h n - H i l l i a r d Equat ion Numerous attempts have been made in recent years to explain the dynamics of phase separation in binary alloys. When a binary alloy, composed of species A and B, is prepared in a state of isothermal equilibrium at a temperature Tj , greater than the critical temperature Tc, the alloy's composition is spatially uniform with the concentration u, of B taking the constant value um. Suppose now that the two component system is quenched (rapidly cooled) to a uniform temperature T2 less that Tc. Then the cooled system will separate itself out into a coexistence of two phases, one phase rich in species A and the other rich in B. A rough description of the behavior of such systems can be provided by consideration of the Gibbs free energy G(u,T), which is single welled for T > Tc, but has the double well form shown in Figure 4.1 for T < Tc. The interval (usa,usb) is called the spinodal interval, where usa and usb are spinodal points and are defined by the conditions < 0 in (usa, usb) and | ^ > 0 outside the interval Near the two local minima are the binodal values ua and ub, which are the two unique tangent points of the curve with the supporting tangent. A state u is said to be stable, metastable or unstable according to whether it corresponds to a homogeneous state which is a global minimizer, local but not global minimizer, or local non-minimizer respectively, for the free energy functional F(u)= I G{u{x),T)dx, (4.1) JQ 90 Chapter 4. Phase Separation Models in One Spatial Dimension 91 at constant average concentrations (i.e., f^udx is constant). For G as in Fig.4.1, the spinodal region usa < u < is unstable, states satisfying u < ua or u > are stable, and the remaining intervals are metastable. A mixture, initially with a homogeneous spatial composition taking values in the spinodal interval, will quickly evolve from this unstable state to an equilibrium configuration consisting of two coexisting phases with a spatial pattern composed of "grains" rich in either A or B. Such an evolution is called phase separation or spinodal decomposition. Then, a coarsening or ripening process ensues on a much slower time scale, as the system losses some of the grains, tending toward more stable patterns. Figure 4.1: Free energy of the system below the critical temperature. A naive attempt to extrapolate the dynamics of phase separation from the energy minimization of (4.1) would lead to a backward-forward heat equation ut = - A Q{u) , with Q(u) = -G'(u). (4.2) Here the diffusion coefficient Q'(u) is positive in the spinodal interval. Thus, the initial value problem is classically ill-posed from the mathematical viewpoint. Experimental Chapter 4. Phase Separation Models in One Spatial Dimension 92 data (cf. [46]) showed the importance of gradient energy effects to the phase separation and coarsening process. One way to obtain a physically justifiable regularization for equation (4.2) is due to Cahn and Hilliard [24] who included the energy contributions in their definition of the free energy. They modified the free energy G(u) of the system by adding a gradient term e\ V u | 2 / 2 to get e 2 mm G(u) = G(u) + ^-\s7u\2 . (4.3) Here e —> 0 is the interfacial energy parameter, G{u) is called the homogeneous free energy and G(u) is known as the Landau-Ginzburg free energy. Given that the mass um is fixed, the Cahn-Hilliard model for the equilibrium description of phase separation is characterized by minimizing the total energy viz. il(u) = J ^G(u) + ^-| V u | 2 | dx , subject to J u(x)dx = um\£l\ . (4.4) The kinetics of phase separation can be modeled using non-equilibrium thermodynam-ics. For an isothermal binary mixture, the mass flux J is proportional to the gradient of the intrinsic chemical potential //, J = - M V M , (4-5) where M > 0 denotes the mobility and the chemical potential fi is the variational deriva-tive of the energy I(u): fi = ^- = -e 2 A u - Q ( u ) . (4.6) bu Then, from the diffusion equation, u t + y • J = 0, assuming the mobility M = 1 , one obtains the Cahn-Hilliard equation ut = A(-e 2 Au-Q(u)) . (4.7) When the viscous stresses arising from the relative fluxes of the two components are taken into account, the diffusion equation as suggested in [78] gets modified to ut = A(fj, + jut), Chapter 4. Phase Separation Models in One Spatial Dimension 93 where the term ^ut represents the viscous effects in the mixture. Taking the chemical potential u as in (4.6), Novick-Cohen [78] obtained the following viscous Cahn-Hilliard equation describing the dynamics of viscous first order phase transitions ut = A(-e2 Au-Q(u) + iut) . (4.8) Here the terms — e2 A 2 u and 7 A ut represent a gradient energy regularization and a viscous stress regularization, respectively, of the ill-posed backward-forward heat equation (4.2). Boundary conditions may be taken to be of Neumann type n • \y(Q(u) — e2 Au + jut) = 0 (no flux) , n • \/u = 0 (variational) , (4.9) where n is the outward unit normal to <90. Another model for phase separation, which was introduced by Rubinstein and Sternberg [85] as a particular limit of the viscous Cahn-Hilliard equation, is the constrained (non-local) Allen-Cahn equation ut = e2 A u + Q(u) / Q(u)dx , xeSl, t > 0 , (4.10a) ^ - = 0, xedtt, t>o. (4.10b) on Note that these phase separation models (4.7), (4.8) and (4.10) can also be recovered from the phase-field equations which arise in the modeling of solidification of super-cooled liquids (cf. [8], [20], [21], [81]) in particular parameter limits. There has been much recent work analyzing the dynamics associated with the phase separation models (4.7), (4.8) and (4.10). These studies have revealed that the phe-nomenon of phase separation and coarsening process in a binary alloy does occur in these models. In one dimensional setting, the dynamics associated with these models typically proceeds in two stages when e is small: a relatively fast stage during which a pattern of internal layers is formed from initial data in an 0(1) time interval, followed by Chapter 4. Phase Separation Models in One Spatial Dimension 94 an exponentially slow coarsening process during which the internal layers move exponen-tially slowly in time until they collapse together in pairs. For the Cahn-Hilliard equation, the existence of metastable internal layer motion has been proved in [2], [9], [15], [17], [32], [76], etc. and an explicit characterization of metastability is given in [15] (see also [35]) using a dynamical system approach. In [88] and [86] an asymptotic projection method is used to obtain similar explicit results for the viscous Cahn-Hilliard equation and the constrained Allen-Cahn equation, respectively. In a multi-dimensional setting, dynamic metastability can also occur for the phase separation models that conserve mass. For the Cahn-Hilliard equation, the motion of radially symmetric internal layer solutions, referred to in [3] as bubble solutions, has been shown to exhibit metastable behavior in [3] and [4]. In [112] the projection method is employed to give an explicit asymptotic description of metastable bubble motion for the constrained Allen-Cahn equation. In this chapter, one of our main goals is to study the similarities and differences of the dynamics of an n-layer (n > 2) metastable pattern associated with the three phase separation models in one spatial dimension mentioned above and to compare our results with those of Bates and Xun [15] and Eyre [35]. The second goal is to use a hybrid algorithm based on our asymptotic information and the conservation of mass condition to characterize the entire coarsening process for these models. To do so, we consider the viscous Cahn-Hilliard equation in the following form (1 - a)ut — -(e2uxx + Q(u) - ctK,ut)xx , -1 < x < 1 , t > 0 , (4.11a) ux(±l,t) = uxxx(±l,t) = 0 ; u(x,0) = u0(x) , (4.11b) where u(x,t) is the concentration of one of the two components in the alloy. Here K > 0 is the viscous parameter, £ —> 0 is the interfacial energy parameter, a is a homotopy parameter satisfying 0 < a < 1, and Q(u) = —G'(u) where G(u) is a double-well potential with wells of equal depth. More specifically, we assume that Q(u) has exactly Chapter 4. Phase Separation Models in One Spatial Dimension 95 three zeros on the interval located at u = S- < 0, u = 0 and u = s+ > 0, with Q'{s±) < 0 , Q'(0) > 0 , G(s±) = 0 . Prototypical is Qiu) = 2(u — u3), for which s± = ± 1 and G(u) = (1 — u2)2/2. When 0 < a < 1, the mass 2um = u(x,t)dx is conserved for (4.11). We assume below that the initial value u0(x) is such that s_ < um < s+ . The importance of the homotopy parameter a is to distinguish three cases: (i) a = 0, (ii) 0 < a < 1 , K 7^  0 and (iii) a = 1 , « 7^  0. In case (i), equation (4.11) reduces to the Cahn-Hilliard equation for spinodal decomposition. In case (iii), we can integrate the right side of (4.11a) twice, explicitly impose a mass constraint, and re-scale r to obtain the constrained Allen-Cahn equation. Case (ii) corresponds to the viscous Cahn-Hilliard equation. Thus, the homotopy parameter a enables us to understand how three different models are related by studying only (4.11). The organization of the chapter is as follows. In §4.2, we describe the asymptotic differential algebraic equations (DAEs) of motion, derived in [88], for the locations of the internal layers corresponding to an n-layer metastable pattern for (4.11), and compare these asymptotic results with corresponding full numerical results. In §4.3, by decoupling this system of DAEs, we study the metastable dynamics associated with the three phase separation models and compare our results for the Cahn-Hilliard equation with those in [15] and [35]. Finally, in §4.4, we propose a hybrid algorithms based on this D A E system and an interface realignment technique to simulate the entire coarsening process associated with these models. 4.2 Dynamics of an n-layer Metastable Pattern In [88], a system of differential-algebraic equations that describes the metastable dynam-ics of an n-layer pattern for (4.11) was derived by applying the projection method. From Chapter 4. Phase Separation Models in One Spatial Dimension 96 this D A E system, the metastable behavior associated with patterns of two internal layers was obtained and the asymptotic results for the layer dynamics were shown to compare very favorably with corresponding full numerical results. However, for a general pattern having n > 2 internal layers, this D A E system was not verified in [88] nor were general properties of the layer dynamics obtained. Here we first present the main asymptotic results in [88] and then supplement it with some numerical results. Let us(z) be the unique heteroclinic orbit connecting s_ and s+, which satisfies u"(z) + Q [us{z)} = 0 , - o o < z < c o ; us(0) = 0; u's(z) = 0; (4.12a) us(z) ~ s+ — a+e~l/+z , z—>+oo ; u„(z) ~ s_ + ci-el/+z , z —> — oo . (4.12b) The positive constant u± and a± in (4.12b) are defined by "± = l-Q'M]" , l o g a ± = l o g ( ± a ± ) + / S ± (TT^V + ——) dV • (4-13) Jo \[2G{T])]I V~ S±J For j = 0 , 1 , . . . , n, we define £j = ( —1) J£ 0 , where £ 0 = ± 1 specifies the orientation of the internal layer closest to x = — 1. For j = 0 , 1 , . . . , n, the triplet (a3,Vj,Sj) is defined by , , (a+,v+iS+) , w h e n ^ = - l , (a3,v3,s3) = < (4.14) (a_,i/_,5_) , when = +1 • Then an n-layer metastable pattern for (4.11) shown in Figure 4.2 is represented by the approximate form u ~ u*(x), where n - 1 u\x) = u*(x;x0,x1,...,xn_1) =u s[£- 1e0(x-a;o)]-|-£ (u«[e _ 1&(a: - x3)} - 5 j ) . (4.15) i=i Here ^ = Zj(*) for j = 0 , l , . . . , n - 1 and x 3 ^ ( t ) < x3(t). Since 1 ^ ( 0 ) = 0 , the curves x — Xi(t), i — 0 , 1 , . . . , n — 1, closely determine the locations of the zeros of u(x,t) during the slow evolution. The internal layer distances dj = dj(t) are given by d3 — Xj — Xj-i for j' — 0 , 1 , . . . , n , where we have introduced the fictitious layers and Chapter 4. Phase Separation Models in One Spatial Dimension 97 xn by x _ i = —2 — xo and x r a = 2 — x „ _ i . The layers are assumed to be well-separated in the sense that d3(t) = 0(1) as e —• 0 for j = 0 , 1 , . . . , n . A lengthy calculation in [88] leads to the following results for the metastable dynamics in the viscous Cahn-Hilliard equation (4.11). 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 L J -1 X _ ! -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 XQ  xl X2 X3 X n - 2 xn-l Xn Figure 4.2: An n-layer metastable pattern for the viscous Cahn-Hilliard equation (4.11) Propos i t i on 4.1 (From [88]) For e —• 0 ; an n-layer metastable •pattern for (4-H) with widely separated layers is represented by (4-15) , where X{(t), i — 0 , 1 , . . . , n — 1, and an unknown function crc(t) satisfy the explicit DAE system n-1 a.Kdz~xx2 + (1 — a) ^2  xkbjk ~ crc£j(s+ — s_) + Hj , j = 0 , 1 , . . . , n — 1 , (4.16a) fc=o n ^T, sk{xk-xk-i) ~ m-en(9--6+). (4.16b) k=0 Here the exponentially weak forces Hj for j = 0 , 1 , . . . , n — 1 and the coupling coefficients Chapter 4. Phase Separation Models in One Spatial Dimension 98 bjk for j, k = 0 , 1 , . . . , n — 1 are defined by H3 = 2 {a)+xv]+xe-t~lv>+li*1 - a ^ e ' * ' 1 ^ ) , (4.17a) bjk = J ^ (u s [£ _ 1 ^.(a; - xk)] - sk) (u s[e _ 1^(a; - Xj)] - s i + i ) dx , (4.17b) where /c o pO p c o [u's(z)]2dz , 9 - = [us(V) - 5_] dV , 9+ = / [s+ - u.{r,)] dV . (4.18) - c o J — oo JO To validate the D A E system (4.16) numerically, we compare the asymptotic and nu-merical results for the internal layer locations Xj corresponding to an n-layer metastable pattern for (4.11) with Q(u) = 2(u — u3) and K = 1. For this form of Q(u), the heteroclinic orbit constants needed in (4.16) and (4.17) can be obtained analytically as a± = 2, i/± = 2, s± = ± l , £ = 4/3, 0 ± = l o g 2 . In the comparisons below, we chose different values of a resulting in different types of phase separation models and took i t (x , 0 ) = u*(x; XQ, . . . , x ^ ^ ) as the initial data for (4.11). Here u* is the n-layer metastable pattern (4.15) and x°} 6 (—1, l ) , j = 0 ,1 , . . . , n — 1 are initial zeros of u. To eliminate any unwanted transient effects, we computed the full numerical solution to (4.11) with these initial data until t = tc, where tc was some positive constant, and reset x ° to be the zeros of the numerical approximation uh at time t = tc. With these new values of x°- as its initial data, the D A E system (4.16) was solved numerically and results were compared with corresponding full numerical results. We solved the D A E system (4.16) using the implicit-ODE solver LSODI (cf. [48]). This solver requires the initial values of crc{t), Xi(t) for i = 0 , . . . , n — 1 and their derivatives. For given initial values of a;,-, for i — 0 , . . . , n — 1, in our computations, crc(0) was determined by the asymptotic estimate (4.25b) or (4.34) below, <rc(0) was set to zero Chapter 4. Phase Separation Models in One Spatial Dimension 99 and ii(0) were obtained by solving (4.16a) which was treated as a linear system of X{. In addition, instead of calculating the coefficients bjk in (4.16a) precisely from (4.17a), we used (4.24) below to evaluate them. To compute numerical solutions to the evolution problem (4.11), we used the trans-verse method of lines approach introduced in §1.3, with a slight modification. Since the right hand side of (4.11) contains the derivative ut, it is convenient to rewrite (4.11) in the form etK.Ut - e 2 u x x + Q(u)- o , rt x.(±l,i) = 0 , (4.19a) ( l - a ) u ( = - o x x , ax{±l,t)=0, (4.19b) with u(x, 0) = uo(x). Then, by replacing the time derivatives ut in (4.19) by the backward differential formulas (BDF), this problem can be converted into a set of boundary value problems with two unknowns un and crn at each time step. These boundary value problems were solved by applying COLSYS [6]. This procedure works for 0 < a < 1, since (4.19) is not well-defined when a = 1 unless a mass constraint is imposed. For the special case a = 1, we computed solutions to the constrained Allen-Cahn equation 1 [ l ut = e 2 u x x + Q(u) - - J ^Q(u)dx , ^ ( ± 1 , ^ = 0, u(x, 0) = u0(x). (4.20) After ut is discretized by the B D F schemes, we obtain 0jUn-j{X) - /?n+l U2(un)xx + Q{un) ~ \ j Q{un)dx\ , (un)x(±i) = 0 , (4.21) j_o V 2 J-i / where k and j3j are same as in §1.3. Since these boundary value problems are non-local, we introduce the new variables z= ( z i , . . . , Z4)1 with zi = u n , z2 = (un)x, z3 = J Q(un)dx, z4 = J ^Q(un)dx/ J ^Q(un)dx . (4.22) With these new variables, (4.21) is reduced to a local problem zx = F{z), 2 2 ( - l ) = z 2 ( l ) = z 4 ( - l ) = 0, ^(1) = 1, (4.23) Chapter 4. Phase Separation Models in One Spatial Dimension 100 where nonlinear function F : R4 —>• R4 can be easily derived from (4.21) and (4.22). Again, COLSYS was used to solve the boundary value problem (4.23) at each time step. It has been clear from [15] and [88] that for the phase separation models specified by (4.11), the closest layers will move towards each other at an extremely slow rate and eventually undergo a collapse phase, leaving behind a metastable pattern with two fewer layers. In Tables 4.1 - 4.3, we give a comparison between the asymptotic and numerical results for the evolution of these collapse layers for three types of phase separation models corresponding to a = 0, | and 1. In these tables, the initial values of Xj for j = 0 , . . . , n — 1 with n = 6 were chosen to be near —0.7, —0.4, —0.1, 0.15, 0.5 and 0.8, so the second and third columns compare the numerical and asymptotic elapsed time necessary for the distance d 3 to be given by the values in the first column. These elapsed times are found to agree to more than three significant digits. In Figure 4.3 - 4.5, we plot the full numerical solutions at different time to (4.11) corresponding to the parameter values used for Table 4.1 - 4.3, respectively. The behavior of the solution during the metastable phase and the collapse phase shown in these figures will be discussed in the next few sections. Figure 4.3: Plot of the numerical solution to the Cahn-Hilliard equation at different times corresponding to the parameter values given in Table 4.1. Chapter 4. Phase Separation Models in One Spatial Dimension 101 d3 t(num.) t(asy.) 0.2499637 0.2490057 0.2457128 0.2353902 0.2221106 0.2049384 0.1503616 0.1028643 0.0697038 0.11805135 x l O 2 0.31421876 x l O 3 0.12120381 x l O 4 0.30076767 x l O 4 0.40610571 x l O 4 0.45628835 x l O 4 0.47902868 x l O 4 0.47961152 x l O 4 0.47963267 x l O 4 0.11805099 x l O 2 0.31423049 x l O 3 0.12121263 x l O 4 0.30080758 x l O 4 0.40615483 x l O 4 0.45634494 x l O 4 0.47909376 x l O 4 0.47967273 x l O 4 0.47971114 x l O 4 Table 4.1: A comparison of the asymptotic and numerical results for t = t(ds) for the Cahn-Hilliard equation (a = 0) with e = 0.03. The initial values of Xj for j = 0 , . . . , 5 were -0.7000000, -0.3999999, -0.0999999,0.1499999, 0.4999999,0.8000000. 4.3 Propert ies of the Metastable Dynamics The D A E system (4.16) provides a quantitative characterization of metastable internal layer motion for the viscous Cahn-Hilliard equation (4.11). However, it is not easily analyzed and thus gives little analytical information about the metastable dynamics unless the system is asymptotically simplified. In this section, we decouple the D A E system (4.16) and then study this reduced system to reveal the analytical behavior of the metastable dynamics associated with the three phase separation models. 4.3.1 S impl i f icat ion of the D A E System (4.16) In [88], the coefficients bjk defined by (4.17b) have been evaluated asymptotically as bjk ~ - ( - l ) i + * ( S i - x f c ) ( 5 + - S _ ) 2 -e ( l - (-1)'"+*) - * _ ) ( * _ - 9+) , for j > k , (4.24) bjj ~ —£tl 5 bjk = 0(e~£ l c ) , for j < k . Chapter 4. Phase Separation Models in One Spatial Dimension 102 ds t(num.) t(asy.) 0.2499080 0.2457898 0.2354383 0.2268495 0.2068120 0.1687788 0.1484624 0.1016572 0.0545736 0.10960657 x l O 2 0.45001770 x l O 3 0.11982005 x l O 4 0.15722918 x l O 4 0.20064226 x l O 4 0.22184970 x l O 4 0.22420676 x l O 4 0.22539916 x l O 4 0.22551925 x l O 4 0.10947517 x l O 2 0.45013814 x l O 3 0.11986815 x l O 4 0.15731235 x l O 4 0.20078002 x l O 4 0.22204742 x l O 4 0.22442599 x l O 4 0.22564691 x l O 4 0.22576482 x l O 4 Table 4.2: A comparison of the asymptotic and numerical results for t = t(d3) for the viscous Cahn-Hilliard equation (a = 0.5) with e = 0.04. The initial values of Xj for j = 0 , . . . , 5 were -0.7000000, -0.3999999, -0.0999996, 0.1499996, 0.5000000,0.8000000. Here u = — us(T])][us(r]) — S-]dr] and c is a positive constant that is proportional to the distance xk — Xj. If we write the left side of (4.16a) in the matrix form Bx, then the matrix B is lower triangular to within exponentially small terms. Moreover, if 0 < a < 1, then the entries in the matrix B that are below the main diagonal are O(e) smaller than the entries along the main diagonal. In this case, a simplified form of the D A E system (4.16) was obtained in [88], although only a preliminary description of the metastable behavior was given there. In addition, numerical verification of the asymptotic results in [88] are needed to make the work in [88] complete. For 0 < a < 1, we follow [88] and introduce the decoupled form of (4.16). Since B is a diagonal matrix to within O(e) terms, the system (4.16) can be asymptotically decoupled for this range of a to obtain n-1 aKtSxj ~ e(pi~1(s+ - 5 _ ) _ 1 J2( s* ~  sk+i)Hk + eHj , j = 0 , . . . , n - 1,(4.25a) k=0 n-1 oc ~ n-1{s+ - s-)-2 J2(sk - sk+1)Hk . (4.25b) k=0 This form is an exact reformulation of (4.16) when a — 1. Let n > 2 and label the Chapter 4. Phase Separation Models in One Spatial Dimension 103 d3 t(num.) t(asy.) 0.2499500 0.2494742 0.2450228 0.2370225 0.2201322 0.2058314 0.1635920 0.1171353 0.0497620 0.11561895 x l O 2 0.12103537 x l O 3 0.10128793 x l O 4 0.21550525 x l O 4 0.34372605 x l O 4 0.39218631 x l O 4 0.43265906 x l O 4 0.43761505 x l O 4 0.43813902 x l O 4 0.11528481 x l O 2 0.12102348 x l O 3 0.10130617 x l O 4 0.21555446 x l O 4 0.34383206 x l O 4 0.39233456 x l O 4 0.43291128 x l O 4 0.43793961 x l O 4 0.43846704 x l O 4 Table 4.3: A comparison of the asymptotic and numerical results for t = t(d3) for the constrained Allen-Cahn equation (a = 1) with e = 0.04. The initial values of Xj for j = 0 , . . . ,5 were-0.7000000,-0.4000000,-0.0999997,0.1499996, 0.5000000,0.8000000. initial layer separations dj(Q) for j = 0 , . . . , n by ctj1 = dj(0). Assume that there is some J with J ^ 0 and J ^ n such that vjd® < t>jd° for all j = 0 , . . . , n and j ^ J. Then, from (4.25a), it is easy to show that the distance dj(t) between xj and xj_i satisfies the approximate evolution equation d f ^ ~ ( l - - ) aWje-'~lvjdj , dj{0) = d°j>0. (4.26) OLKp \ nj Integrating (4.26), we obtain dj{t) ~d° + - log[l - , * . ^ a ^ [ l - 2 l n ] ~ \ ^ . (4.27) Thus, dj = 0(e) when t w ts. Since the right hand sides of (4.25a) do not depend on a, it is clear that the viscous Cahn-Hilliard equation (0 < a < 1 ) has the same metastable dynamics as the constrained Allen-Cahn equation (a — 1 ) except for the scale of the collapse time. For the Cahn-Hilliard equation (a = 0), an asymptotic simplification of (4.16) is not so straightforward. Using (4.24), adding up two consecutive ODEs in (4.16a), differentiating Chapter 4. Phase Separation Models in One Spatial Dimension 104 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 4.4: Plot of the numerical solution to the viscous Cahn-Hilliard equation at dif-ferent times corresponding to the parameter values given in Table 4.2. (4.16b) and performing certain rescalings, we can obtain ^ 1 - e/i 0 -epo 0 0 — 1 1 — epi —epi 0 -1 1 - eu2 -ep2 0 0 0 0 0 0 - I f " ( - I ) " (-1)" • • • l - e / * » - 2 SPn-2 1 -1 -1 - i f - 1 (-iy x0 XX x2 xn-2 y %n-l J where y,k and rk are defined by, for 0 < k < n - 2, Pk = rk = (Xk+l - xk)(s+ - s _ ) 2 1 ixk+l - xk)(s+ - 5 _ ) 2 2 (Xk+l - xk)(s+ - 5 _ ) 2 r0 r2 rn-2 \ 0 / (4.28) (4.29a) (Hk + Hk+1) dk+* - a \ v \ e - e lukdk) . (4.29b) From the first three rows of (4.28), it is not hard to derive that x0 ~ efiQxx + r0 , xt, ~ e/.txx2 + r 0 + rx and x2 ~ ef.i2x3 + rx + r2 + efix(r0 + rx). In general, we can use the Chapter 4. Phase Separation Models in One Spatial Dimension 105 0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 4.5: Plot of the numerical solution to the constrained Allen-Cahn equation at different times corresponding to the parameter values given in Table 4.3. induction principle to show that 3 bj ~ e N x ] + 1 + £ £J~k (rk-i + rk) j j f*i, for 1 < j < n - 2 . (4.30) k=\ l=k Multiplying the last row of (4.28) by (—l) n _ 1 and adding it to the second to last row, we have -eun-2xn-2 + (1 - enn-2)xn-i ~ r n _ 2 , (4-31) which can be combined with (4.30) to yield i n - i ~ X>n_2-fcr* II u, • (4.32) k=0 l=k+l Substituting (4.32) into (4.30), we can obtain x3 for j — n — 2 , . . . , 0 , recursively. Specif-ically, the internal layer locations Xj(t) for j = 0 , . . . , n — 1 satisfy n - 2 fe-i j - i i - i X3 ~ k=3 1=3 k=0 l=k+l (4.33) where j . i k and rk are given in (4.29). From (4.16a) and (4.33) with j — 0, it is straight-forward to get the following estimate for crc(t) •^)(*+e"SeSsW- (4-34) Chapter 4. Phase Separation Models in One Spatial Dimension 106 Since urk — /.ik(Hk + #fc+i) , we can neglect some higher order terms in this estimate to obtain o-c ~ - T T T ^ - T n/*/ • (4.35) t;o(s+ - s-) k = 0 l = Q As a partial check on our simplified results, the estimates for ac in (4.25b), (4.34) and (4.35) were evaluated for various sets of internal layer locations and were found to compare very favorably with corresponding numerical results computed directly from (4.16) using LSODI. The results are shown in Table 4.4 and 4.5. (x0,x1,x2,x3) <7C (num.) o-c (4.34) <TC (4.35) (-0.9,-0.4,0,0.6) (-0.4,-0.1,0.1,0.5) (-0.5,-0.2,0.2,0.8) (-0.6,-0.2,0.2,0.6) (-0.7,-0.3,0.25,0.6) 0.336661 x IO" 7 -0.117992 x 10~8 -0.149581 x IO" 1 1 -0.697625 x IO" 1 6 -0.729543 x 10~1 6 0.336380 x IO" 7 -0.109940 x IO" 8 -0.149597 x IO""11 -0.696720 x IO" 1 6 -0.725283 x IO" 1 6 0.329785 x 10~7 -0.104577 x IO" 8 -0.144732 x 10~ n -0.679726 x 10~1 6 -0.707289 x IO" 1 6 Table 4.4: A comparison of the asymptotic and numerical results for ac for the Cahn-Hilliard equation (a = 0) with e - 0.02 and Q(u) = 2(u - u3). Here n = 4 and o^ = l . (xo,xi,x2,x3) <Tc (num.) <TC (4.25b) (-0.5,-0.1,0.1,0.5) (-0.7,-0.3,0,0.5) (-0.5,-0.1,0.3,0.7) (-0.75,-0.25,0.2,0.75) 0.127273 x I O - 4 0.162085 x IO" 7 -0.213520 x IO" 1 0 0.738668 x IO" 1 2 0.129567 x 10 - 4 0.164682 x 10" 7 -0.209847 x IO" 1 0 0.747657 x IO" 1 2 Table 4.5: A comparison of the asymptotic and numerical results for ac for the viscous Cahn-Hilliard equation (a = |) with e = 0.03 and Q(u) = 2(u - u3). Here n = 4 and 6 = 1-Chapter 4. Phase Separation Models in One Spatial Dimension 107 4.3.2 Comparisons of the Internal Layer Dynamics Using the decoupled O D E systems (4.25) and (4.33), we are able to study analytically the metastable behavior associated with the viscous Cahn-Hilliard equation (4.11). We first consider the Cahn-Hilliard equation (a = 0). Let n > 4 in (4.33) and label the initial layer separation dj(0) for j = 0,... ,n by a*j = dj(0) . Assume that for some J with 2 < J < n — 2, we have that vjd® < Vjd°- for all j = 0 , . . . , n and j ^ J. Then, from (4.29b), it is easy to show that and the rest of the ?-j's are exponentially small, compared with r j _ 2 and rj. Now, from (4.33), we have . x j _ 2 ~ r j _ 2 > 0 , x j _ i ~ r j _ 2 > 0 , xj ~ rj < 0 , xJ+1 ~ rj < 0 . (4.37) This means that the interfaces of the annihilating interval (XJ-I,XJ) will approach each other and eventually disappear, while the nearest neighboring intervals move together in an asymptotically rigid way. For the layers that are left of x j _ 2 and right of x j + i , we can find from (4.33) that J-3 xj-k ~ e * - 2 r j_ 2 [ J m ~ 0(£k-2e-£~ll/jdj) > 0 , k > 2 , (4.38a) l=J-k J+k-1 xJ+k ~ £ f c -V j T] ^ ~ -0(e f c- 1e- e" 1"^-') < 0 , A; > 1 . (4.38b) Therefore, we expect that the layers x j _ x and x j _ 2 ( x j and x j + i ) will move at the same speed and direction to the right (left), and that the other layers on the left (right) will move in the same direction as X j _ i and x j _ 2 ( x j and x j + i ) , but at a successive slower speed, i.e., xj_kl with k\ > 3 (xj+k2 with k2 > 2) is 0(e) slower than its right neighbor x j _ f c 1 + i (left neighbor xj+k2-i)- This analysis has been verified numerically (e.g., see Chapter 4. Phase Separation Models in One Spatial Dimension 108 Figure 4.3). From (4.36) and (4.37), it is easy to show that the distance dj(t) between xj and xj_i satisfies d'j ~ — 2——a2Ju2Je-£-1^ (dji, + , dj(0) = d°j > 0 . (4.39) (S+ — 5_J Since the distances dj_x and are constant to within 0(e) terms before the layer collapse, we integrate (4.39) to obtain n e e ( s + - s_)2 (d0, A1 + d 0 - 1 ) ' 1 . J 0 4 , ( 0 ~ <# + - l o g ( 1 - * / * . ) , * . = — ^ i z l _ £ ± l _ i _ c . - > , , 4 (4.40) Here r s is the approximate collapse time for the Cahn-Hilliard equation. We now return to study the motion of the layers characterized by the ODE system (4.25a) for the viscous Cahn-Hilliard equation with 0 < a < 1. We note that in this case the metastable dynamics is insensitive to the value of a except for the time scale of the motion. Let n > 3 and assume that there is some J with 1 < J < n — 1 such that vjd°j < u3dQ- for all j = 0 , . . . , n and j ^ J. Then, from (4.17a), we have Hj ~ —Hj_i ~ — 2a2i/je~e  l l / J d J , while the other i/j's are exponentially small compared with Hj and . Thus, using (4.25a), it is easy to obtain that andxj-x ~ e#j_i[l - 2/n] > 0 ; andij ~ - e ^ j _ i [ l - 2/n] < 0 ; (4.41a) a«/?i:j_i ~ i > 2 ; a K / 3 i J + i ~ — ( - l ^ j - x , i > 1. (4.41b) Therefore, when 0 < a < 1, as the collapse layers move toward each other, the other layers move asymptotically at speeds of a same order as the collapse layers and in alternate left and right directions (see Figure 4.4 and 4.5). In other words, the mass of the disappearing "island" will be consumed evenly by the rest of the "islands" by expanding their widths. This has been verified by the full numerical results in Figure 4.4 - 4.6. Now let's analyze the motion of internal layers with collapse layers close to the bound-aries. For the Cahn-Hilliard equation, let n > 3 and assume that for J = 0, l , n — l o r n , Chapter 4. Phase Separation Models in One Spatial Dimension 109 we have that vjd°j < Vjd°- for all j = 0 , . . . , n and j / J. If J = n - 1, then 2 n~ 3 dn_2(s+~sY n-^ e will be dominant, as the other r/s are exponentially small compared with rn_3. Thus, we obtain from (4.33) that n—4 ij ~ e " - 3 - V n _ 3 TT ^  > 0 , for 0 < j < n - 4, (4.42a) 1=3 xn-3 ~ r n _ 3 > 0 , ai n_ 2 ~ r n _ 3 > 0 , i n _ i ~ £ ^ n _ 2 r n _ 3 > 0 . (4.42b) Using these asymptotic estimates, it follows that all layers will move to the right, but their speeds are different: while the layers xn_3 and xn-2 are shifting at the same speed rn_3 , other layers are almost static. This is verified in Figure 4.6(a). In Figure 4.6-4.8, we plot the full numerical solutions to the viscous Cahn-Hilliard equation (4.11) for the metastable phase in parts (a), (c) and (e) and for the collapse phase in parts (b), (d) and (f). In each of these figures, we plot the solutions at different times for (4.11) corresponding to a = 0, a = | and a = 1. From (4.42b), we have that L - i ~ - . , 2 ^ - i ^ - i e -^ " - 1 ^- 1 , (4-43) and dn-2 is constant to within 0(e) terms before the annihilation. Hence, we can integrate (4.43) to obtain (4.44) dn-i(t) ~ C i + — log (1 -</*.) , ts =  £ { s + o 2 3 - ^ / 0 - 2 e ' - 1 ^ - ^ - , . Next, consider the Cahn-Hilliard equation with J = n. Then 9 d n _ i ( 5 + - s_y will be dominant in this case. From (4.33), it is easy to show that n—3 XJ ~ e n ~ 2 - - ? r n _ 2 J] m > 0 , for 0 < j < n - 3 , (4.45a) 1=3 xn-2 ~ rn_2 > 0, ~ r n _ 2 > 0. (4.45b) Chapter 4. Phase Separation Models in One Spatial Dimension 110 Thus, the layers xn_2 and xn-\ will move to the right at the same speed r n _ 2 , while other layers will move in the same direction, but at a much slower speed. This is verified in Figure 4.7(a). In a similar way as in the derivation of (4.44), we can estimate the length of the annihilating interval (x n,a; n_i) as 4i (0 ~ <£ + ^ l ° g ( 1 - * / * . ) , ^ ^ T ^ ^ 4 - (4-46) For J — 0 or J = 1, it is clear by symmetry that the metastable behavior of a pattern of n internal layers is similar to that for the case J = n or J = n — 1, respectively. For the viscous Cahn-Hilliard equation (4.11) with 0 < a < l , l e t n > 2 and assume that for J = 0 or J = n, we have that vjdj < Vjd°- for all j = 0 , . . . ,rc and j ^ J. When J = n , we have from (4.17a) that Hn-i ~ 1a\v\ e~£~'Lundn and that other Hj's are exponentially small compared with Hn-\ . Thus, (4.25a) reduces to CLKBX^ ~ e(l - - ) # n _ i , CLK&XJ ~ - ( - l ) n _ i / r n _ 1 , for 0 < j < n - 2. (4.47) n n This means that x n _ i will collapse at the boundary and other layers will be shifting at the same speed and at opposite directions. As a results, the change of the mass clue to the annihilation of xn-\ will be compensated by the remaining layers evenly. This is verified in Figure 4.7(c) and (e). Using dn = — 2xn_\ and (4.47), we have 4.(0 ~ < + - l o g ( 1 - * / « . ) , t,^^lLzlMle^<. (4.48) The dynamics is similar when J = 0. Next, we consider the dynamics of an n-layer metastable pattern which has two smallest neighboring "islands". Specifically, we assume that n > 4 and there is some J with 2 < J < n — 2 such that vjdj = isj+idJ+1 < Vjd°- for all j = 0 , . . . , n and j J, J + 1.. For the sake of simplicity, we let Q(u) be an odd nonlinearity for which a+ — ci- = a and v+ = i/_ = v . In this case, d° = and thus we further assume that Chapter 4. Phase Separation Models in One Spatial Dimension 111 d ~ dj ~ dj+i as e —•> 0 during the metastable phase. Defining H = 2a 2v 2 e~6 l v d , we obtain from (4.16) that Hj_i ~ H and i / j + i ~ — i 7 , and that the remaining rYj's are asymptotically negligible. From now on in this section, we use the notation / = o'(g), by which we mean that / is exponentially small with respect to g as e —> 0. For the viscous Cahn-Hilliard equation with 0 < a < 1, we can find from (4.25) that ac = o'(H) and dKi3xj-i ~ e i / j - i > 0 , O K ^ i j + i ~ £-ff/+i < 0 , x3 = o '( if) , (4.49) for j ^ J — 1, J + 1. Thus, we can claim that during the metastable phase, the layers x j _ i and xj+\ move towards each other, while the other layers, including the interface xj joining the smallest intervals (xj-i,xj) and (xj, X j + i ) , remains stationary in time to within exponentially small precision. For the Cahn-Hilliard equation (a — 0), it is easy to show from (4.29b) and (4.33) that 1 1 rj-2 ~ -, -. To Hj-x , r j _ i ~ — — rYj_! , d j _ i ( 5 + - s_y dj(s+-S-Y 1 1 r3 = o'{H) , for j ^ J± 1, J ± 2 . Therefore, the internal layer locations Xj(t) satisfy ij-2 ~ 0 - 2 > 0 , x j - i ~ r j _ i + r j _ 2 > 0 , ~ r J + 1 < 0 , xj+i ~ r j + r / + 1 < 0, (4.51) x , = 0(e#), for jfi J ± 1, J ± 2 . Here we have to assume G ? J _ ! = <ij+2 to ensure that our former assumption dj ~ <ij+i is valid. Therefore, the internal layers whose motion is most noticeable during the metastable phase are x j _ 2 , that move to the right and x j + i , xj+2 that move to the left. The other layers remain stationary to within at least 0(e) precision. In ad-dition, from (4.51), it is clear that xj±i moves at a higher speed than does xj±2. The Chapter 4. Phase Separation Models in One Spatial Dimension 112 numerical evidence supporting the analysis given above can be observed in Figure 4.8(a), (c) and (e). 4.3.3 Other Explicit ODE Systems There are some other systems of ODEs for the motion of the internal layers for the Cahn-Hilliard equation (cf. [15] and [35]) and it is of interest to determine if these seemingly different systems are consistent with (4.16). In our notation, the ODE system (4.36) in [15] can be rewritten as x i ~ r i +  rj-i J for 1 < < n - 2 , x0 ~ r 0 , x n - i ~ r „_ 2 , (4.52) where r'}s are defined in (4.29b). Comparing this system with (4.33), we find that if the "higher order terms" in (4.33) (i.e., EkZj+i ek~3rk n?"/ u, + Ei'Jo e3'1'^, nfc*+i /•*') are omitted, then our O D E system (4.33) reduces to (4.52). However, it is clear that these "higher order terms" may be significantly greater than the remaining r^s. For example, suppose J with 2 < J < n — 2 (n > 4) is the index of a unique annihilating interval. Then our asymptotic formula (4.38) indicates that the internal layer Xj for j — J — k with k > 2 and j = J + k with k > 1 will move at an algebraic slower speed than the annihilating layers. On the other hand, from (4.52), the corresponding internal layers Xj will satisfy Xj = o'(xj), i.e., they move exponentially slower than the collapsing layers. Thus, the results in [15] provide useful information for the motion of layers for the annihilating interval and its two nearest neighborhoods, but may be inaccurate for other layers. Similarly, for the case corresponding to Figure 4.7(a) and (b), where the annihilating interval is ( x n _ i , . T „ ) , the system (4.52) yields x'j = o'(rn_i) for j < n — 3. In contrast, our ODE system (4.33) gives (4.45a) instead. Here we illustrate that (4.45a) is correct by giving the full numerical results for the locations Xj, their deviations e3 = Xj — x°- and the ratios p3 = e j / e J + 1 at two different times in Chapter 4. Phase Separation Models in One Spatial Dimension 113 Table 4.6. From (4.45), we expect that the ratio p3 with j < n - 3 should be 0(e), and the deviation e3 should satisfy 0 ( e n _ 2 _ J e„_ 2) for j < n — 3. This is exactly what is obtained in Table 4.6. Eyre [35] also derived an explicit O D E system using a collocation technique, but we can't find any similarities between his* results and our asymptotic and full numerical results. t = = 140.22 t = : 156.02 j Xj e i PJ Xj ej Pj 0 -.800000 .000000 - -.800000 .000000 1 -.499998 .000002 .05 -.499995 .000005 .04 2 -.099961 .000039 .05 -.099861 .000139 .05 3 .200718 .000718 .04 .202578 .002578 .04 4 .617812 .017812 1.04 .671481 .071481 .81 5 .917114 .017114 .987852 .087852 Table 4.6: Numerical results (using TMOL) for the Cahn-Hilliard equation (a = 0) at two different times corresponding to the parameter values used for Figure 4.7(a) and (b). Here, x3 for j = 0 , . . . ,5 are the locations of internal layers, e3 = Xj - x°j and Pj = e:/ej+i-4.4 S imulat ion of the Ent i re Coarsening Process The D A E system (4.16) is not valid when two internal layers, or an internal layer and a wall, become closely separately by an amount of 0(e). In particular, when two approach-ing internal layers become closely separated, the layers will undergo a strong local inter-action which leads to an annihilation of two internal layers, leaving behind a metastable pattern with two fewer layers. This strong local interaction of two approaching inter-nal layers during annihilation is very complicated (see Figure 4.3-4.8) and will not be discussed. Instead, we will find an approximation of the interface realignment after an Chapter 4. Phase Separation Models in One Spatial Dimension 114 annihilation, using the explicit characterization of motion of the layers in the previous section. Here interface realignment refers to an algorithm that maps interface locations before an annihilation to locations immediately after an annihilation. Since an annihila-tion event takes place on a much faster time scale than metastable dynamics, we believe that incorporating the interface realignment into the metastable evolution described by the D A E system (4.16) could provide, approximately, a complete quantitative description of the coarsening process associated with the phase separation models. In our discussion below, we assume that at any time, the interval having the least length is unique, i.e., there is some J with 0 < J < n such that vjdj < Vjdj for all j = 0 , . . . , n and j ^ J . Another assumption is that the solution u(x:t) is a piecewise constant with u = s+ or u = s_ during the metastable phase. This is reasonable when e is small since the interfaces of the metastable pattern (4.15) have length 0(e). Under this assumption, the mass constraint is equivalent to the length conservation conditions that both £ \ Q J J d{ and ^ even^t are constant. Here and only here the interval lengths do and dn represent .To + 1 and 1 — xn-\ , respectively. Since the interval with the least length annihilates first, the layers xj_x and xj will be referred to as the left and right annihilating interfaces, while the layers xj_2 and will be referred to as the left and right nearest interfaces. During the annihilation of the J-th interval, the interface number decreases by two (one if J = 0 or n). Let the locations of the resulting interfaces immediately after the annihilation be denoted by x'- for j — 0 , . . . , J — 2, J + 1, . . . , n — 1, where we assume that the set {0,.. . , J — 2} ({J + 1,. . . , n — 1}) is empty if J < 2 (J > n - 2). We now describe how the interface realignment is realized. First, consider the viscous Cahn-Hilliard equation with 0 < a < 1 , and assume J ( ^ 0,n) is the index of the only annihilating interval. It has been found in §4.3 that the annihilating interfaces approach each other and eventually coalesce and that the other interfaces move at the same speed Chapter 4. Phase Separation Models in One Spatial Dimension 115 and in opposite directions. So the new locations of the internal layers after an annihilation can be approximated by x ' ^ X j + i - i y - ' S , j = 0 , . . . , n - l , j ^ J - l , J , (4.53) where 8 > 0 is a constant. Using the length conservation conditions, we have 8 — djl(n — 2). If J = 0 or n, then the new internal layer locations after an annihilation will approximately be x'j = xj + ( - i y - J 6 , j = 0 , . . . , n - l , j ^ ( l - I ) J , (4.54) with 8 = dj/2(n - 1). For the Cahn-Hilliard equation, the asymptotic analysis and numerical experiments in §4.3 indicate that when 2 < J < n — 2 (n > 4), the nearest interfaces xj_2 and xj+i will move at the same speeds and directions as the corresponding annihilating interfaces xj-i and xj, while all other interfaces remain unchanged to within at least 0(e) precision. Thus, the new locations of the interfaces can be approximately represented by x'j_2 = x j _ 2 + 8dj , x'J+1 = x'j_2 + + dJ+1 , (4.55a) x'j = XJ , j = 0 , . . . , J - 3, J + 2 , . . . , n - 1 . (4.55b) Here 8 > 0 is a constant to be determined. To find an approximation for 8, we compare the speeds of the nearest interfaces during the metastable phase. It is clear from (4.37) that the annihilating interfaces move towards each other, each being followed by its nearest interface moving at approximately the same speed. The ratio of the speeds of these two rigid motions is equal to the inverse ratio of the distances between the pairs of the nearest and annihilating interfaces. So, we obtain 8 = d j + 1 . dj-i + dj+i Chapter 4. Phase Separation Models in One Spatial Dimension 116 Note that if dj_i = f/j+i, then 8 = | and the length of the annihilated interval is redistributed equally to its neighboring intervals. The above arguments are limited to the annihilating intervals that are separated from the boundary by at least two intervals. When the annihilating interfaces are near the boundary, it is not hard to see from (4.42) and (4.45) that if J = n — 1 or n, then the interface(s) Xj for J — 1 < j < n — 1 will eventually disappear. Since, during this annihilation, the interfaces x3 for 0 < j < J — 2 are almost unchanged, the new location of the left nearest interface x j _ 2 can be calculated using the length conservation conditions. Specifically, we have x'n_3 = xn_i - dn_2 , x'j = x3 , for j — 0 , . . . , n — 4 , if J = n - 1, (4.56a) x'n_2 = 1 — d n _i , x'j = Xj, for j = 0 , . . . , n — 3 , if J = n . (4.56b) The interface realignment for J = 0 or 1 can be implemented similarly. Our discussions here are motivated by the work of Eyre (see [35]) for the Cahn-Hilliard equation. Com-paring (4.55) and (4.56) with the corresponding equations for the interface motion during annihilation in [35], we find they are essentially equivalent. We now present a procedure, that is conceived to be able to approximately describe the entire coarsening process associated with the phase separation models. We first integrate the D A E system (4.16) until a collapse criterion is satisfied. Then we use the interface realignment technique to determine the new locations of the interfaces after an annihilation, and with these new interface locations as initial values, we return to integrate the D A E system again. We repeat the procedure above for each successive collapse event until a stable equilibrium state with only one internal layer is achieved. Using this procedure, we calculate and plot the interface locations as a function of time for the Cahn-Hilliard equation and the constrained Allen-Cahn equation in Figure 4.9. This figure shows the entire coarsening process associated with the phase separation models and the two distinct time scales of the fast annihilation and the metastable interface Chapter 4. Phase Separation Models in One Spatial Dimension 117 motion. Next, we explain a prominent phenomenon that we can observe from Figure 4.3-4.8. These figures show that for the viscous Cahn-Hilliard equation with 0 < a < 1, the tops of the non-collapse "islands" and the bottoms of the non-collapse "valleys" will deviate some visible distance vertically from the original positions u ~ s± . This deviation, however, does not occur for the Cahn-Hilliard equation (a = 0) during the coarsening process. By examining the version (4.19) of the viscous Cahn-Hilliard equation, we conjecture that this behavior may result from the different signs and values of a(x,t) for different models. Specifically, we can expect from the graph for Q(u) that if a < 0 (<r > 0), then the top of an "island" and the bottom of an "valley" will go up (down), and furthermore, the deviated distance will depend on the value of a. Now we derive asymptotic expressions for a(x,t) for different models to interpret this difference. It has been shown in [88] that we can decompose cr(x,t) as n - 1 a(x, t) = (1 - a) E XjMj(x\x3) + ac(t) , (4.57) j=0 where Mj defined by Mj = $Z\ ( ' u * [s _ 1 ^j(» — Xj)] — s3)dn satisfies Mj ~ 0 i f K i j , M3 ~ £j(s+ - s-){x - XJ) + e{9- - 9+) if x > x3, (4.58) and o~c(t) is asymptotically determined by (4.16). For the viscous Cahn-Hilliard equation with 0 < a < 1, l e t n > 3 and assume that there is some J with 1 < J < n — 1 such that vjdj < Vjd°- for all j. = 0 , . . . , n and j ^ J. Then, the dominant i/j's can be found from (4.17a) to be Hj-x ~ 2ajv) e - € ' l y j d j = H and Hj ~ - H . (4.59) Thus, using (4.25b), we can estimate ac as ac ~2Zjn-1(s+ - s ^ H . Chapter 4. Phase Separation Models in One Spatial Dimension 118 For x < xQ, it is obvious from (4.57) and (4.58) that a ~ ac. For x > x0, we calculate a(x,t), using (4.41), (4.57) and (4.58), to obtain <7 = <7c[l + ( l - a ) 0 ( e ) ] . Therefore, for the constrained Allen-Cahn equation (a = 1), a = ac(t) is constant and its sign depends only on £j . If £j < 0, then the non-collapse "islands" and "valleys" will shift up, and otherwise, they will shift down. For the viscous Cahn-Hilliard equation with 0 < a < 1, a(x,t) differs from ac(t) by only a relative error 0(e) and thus, the movement of these "islands" and "valleys" are the same as for the constrained Allen-Cahn equation. This analysis agrees with the numerical results shown in Figures 4.4-4.8. For example, in Figures 4.4 and 4.5, we have that £ j = 1 with J = 3 and ac is positive. Thus, the metastable pattern on the non-collapse intervals seems to be lifted up during the coarsening process. For the Cahn-Hilliard equation, let n > 4 and assume that for some J with 2 < J < n — 2, we have vjdj < v^d® for all j = 0 , . . . , n and j ^ J. Then, the dominant Hfs are also and Hj satisfying (4.59). From (4.35), we have ~ - - — ^ r e ^ U -euj-x) II (4-6°) For X k - i < x < Xk with 0 < A: < J — 1, we can derive from (4.38a), (4.57) and (4.58) that fc-l J-2 j=o l=j which gives a ~ - i k H e ^ ^ C . (4.61) where C > 0 is a constant. For xk-i < x < Xk with J — l < f c < J + l , i t can be similarly obtained that a ~ £j H C, for some C > 0, which yields that the sign of a Chapter 4. Phase Separation Models in One Spatial Dimension 119 determined only by £ j is the same on the collapse interval and its two nearest neighbors. By symmetry, when xk-i <x<xk,J-rl<k<n,a can be naturally expected to satisfy a ~ - Z k H e k - J - 1 C , (4.62) where C > 0. We notice from (4.61) and (4.62) that the sign of a changes when x crosses the interface xk (k ^ </—1, J). More specifically, if £ k = 1 ( — 1) with k ^ J , which means that u ~ 3_ (s + ) in the fc-th interval (xk-x,xk), then since a is negative (positive) on this interval, the bottom (top) of the corresponding "valley" ("island") will move towards u — 0 during the metastable phase and then move back to its original position at the end of the collapse phase. Therefore, the metastable pattern will contract vertically and will not go beyond u — — 1 and u = 1. In addition, from (4.61) and (4.62), we believe that during the collapse phase, a will be 0(1) only on the collapse interval and its left and right nearest neighbors, but will be at most 0(e3~k~x) (0(ek~J~1)) on the fc-th interval with k < J — 1 (k > J + 1). So we anticipate that the changes of these tops and bottoms are virtually indistinguishable when they are far away from the collapse interval. The full numerical results in Figures 4.3, 4.6 and 4.7 show that the solution u(x,t) to the Cahn-Hilliard equation satisfies — 1 < u < 1 and the local maximums or minimums on all intervals except the collapse interval and its nearest neighbor(s) are almost unchanged during both the metastable phase and the collapse phase. Chapter 4. Phase Separation Models in One Spatial Dimension 120 (e): a = 1, e = 0.04 (f): a = 1, e = 0.04 Figure 4.6: Plots of the full numerical solutions (using T M O L ) to the viscous Cahn-Hilliard equation (4.11) with Q(u) = 2(u - u3) and K = 1 at different times for various values of ct and e. Here the initial data u(x,0) EE u*(x; a?0) with x° — ( 0 8 -0.5, -0 .1 , 0.2, 0.6, 0.8) and u* defined by (4.15). Chapter 4. Phase Separation Models in One Spatial Dimension 121 1=0 t=130.6 t=1S2.0 1=156.05 -0.8 -0.6 -0.4 -0.2 (a): a = 0, e = 0.03 t=156.05 1=156.08 t=156.10 1=156.1  -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 (b): a = 0, e = 0.03 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 (c): a = 0.5, e = 0.04 -0.4 -0.2 0 0.2 0.4 0.6 0.8 (d): a = 0.5, e = 0.04 0.2 0.4 0.6 (e): a = 1, e = 0.04 (f): a = 1, e = 0.04 Figure 4.7: Plots of the full numerical solutions (using T M O L ) to the viscous Cahn-Hilliard equation (4.11) with Q(u) = 2(u - u3) and « = 1 at different times for various values of a and e. Here the initial data u(x,0) = u*(x; x°) with x° = (-0.8, -0.5, -0 .1 , 0.2, 0.6, 0.9) and defined by (4.15). Chapter 4. Phase Separation Models in One Spatial Dimension 122 1391.2 1400.3 1=140.4 t=1400.5 (a): a = 0, e = 0.032 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 (b): a = 0, e = 0.032 t=0 1=66.1 t=1220.3 t=1495.9 -1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 1 (c): a = 0.5, e = 0.042 (d): a = 0.5, e = 0.042 (e): a = 1, e = 0.045 (f): a = 1, £ = 0.045 Figure 4.8: Plots of the full numerical solutions (using T M O L ) to the viscous Cahn-Hilliard equation (4.11) with Q(u) = 2(u - u3) and K = 1 at different times for various values of a and e. Here the initial data u(x,0) = u*(x; x°) with a; 0 = (-0.8, -0.4, -0 .1 , 0.15, 0.4, 0.7) and u* defined by (4.15). Chapter 4. Phase Separation Models in One Spatial Dimension 123 (b) a = 1 Figure 4.9: Plots of the interface locations as a function of time for (4.11) with e = 0.02, K = 1 and Q(u) = 2(u — u3). Here the initial locations of the interfaces are -0.8, -0.6, -0.5, -0.2, 0.05, 0.4, 0.66, 0.8, and the final stable equilibrium having one interface at x0 = -0.01 is achieved at t ~ .2096 x 102 1 for a = 0 and * ~ .9551 x 10 1 6 for a = 1. Chapter 5 Numerical Analysis of an Exponentially Ill-Conditioned BVP In this chapter we give a preliminary approach for the numerical analysis of an expo-nentially ill-conditioned boundary value problem. Although we do not study how to overcome the difficulties in numerical computations of singularly perturbed problems exhibiting dynamic metastability in a general setting, the detailed analysis of a particu-lar example shows the numerical difficulties encountered in computing these metastable problems. 5 . 1 Introduction There have been numerous computational experiments of exponentially ill-conditioned problems such as the viscous shock problem, the exit problem and various phase separa-tion models (cf. [32], [33], [76], [62], [67], [9], [8], [86], [87], [88], etc.). However, little is known of the rigorous nature concerning the convergence and stability of the numerical schemes that compute metastable behavior. As shown in the previous sections, the solu-tion u(x) of an exponentially ill-conditioned boundary value problem, say LEu = / , can be exponentially sensitive to all the data in the equation; for example, to the right-hand side / . Suppose L e is the linearization operator of L e and Ao is its principal eigenvalue which is usually exponentially small. Then, a perturbation A / to / may cause an ex-ponentially large change A u in u, of the order 0(Ao 1 A / ) . Since a good discretization scheme often inherits the properties of the corresponding continuous problem such as the sensitivity and stability, it is natural to expect that a truncation error which is usually 124 Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 125 much greater than the exponentially small eigenvalue A 0 may result in such a large error in the numerical solution uh(x) that renders uh(x) to be highly inaccurate. To overcome this obstacle, in consequence, one has to require that the truncation error be smaller than the smallest eigenvalue Ao, which is exponentially small. Thus high-order or spectral-type numerical methods are preferred for solving these problems. A high-order integral equa-tion scheme was implemented to compute boundary value resonance solutions in [67] and a Galerkin spectral method was employed to solve the Cahn-Hilliard equation in [9] and the viscous Cahn-Hilliard equation in [8]. Another approach to overcome the difficulties in solving exponentially ill-conditioned problems is preconditioning; for example, the vis-cous shock problem and the (constrained) Allen-Cahn equation were successfully studied numerically using a W K B formulation which leads to well-conditioned problems in [87] and [86], respectively. On the other hand, we notice that many conventional schemes have also been widely used in computing solutions to exponentially ill-conditioned problems and they indeed work rather successfully. Elliott and French [32] studied the metastable dynamics for the Cahn-Hilliard Equation by applying the Galerkin finite element method; the classic finite difference schemes were employed to compute the solutions of the Burgers' equation in [62] and the viscous Cahn-Hilliard equation in [88]; Carr and Pego [26] obtained very nice pictures of the evolution of the solutions to the unconstrained Allen-Cahn equation by using the subroutine LSODI with the methods of lines, taking 801 grid points. The success of application of these traditional numerical methods with moderate mesh sizes gives rise to some queries about the heuristic inference in the previous paragraph. Specifically, is it a necessary condition for a numerical method when solving a very i l l -conditioned problem that the truncation error be less than the order of the principal eigenvalue Ao of the corresponding linearized elliptic operator? Can we apply the classic finite difference schemes and other numerical methods with moderate mesh sizes to solve Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 126 •the ill-conditioned problems, instead of using high-order methods or preconditioning? If so, what difficulties will possibly occur in our computations and which type of schemes may be preferred for these problems? Moreover, can we give a rigorous proof of the convergence of a numerical method for an exponentially ill-conditioned problem? Our goal is to shed some light on the above questions and provide some general guid-ance or principles in designing numerical schemes for metastable problems by studying the following singularly perturbed boundary value resonance problem (cf. [1], [31], [60], [67], [74], [113]) Ltu = - e u x x 4- x2m+1p(x)ux = 0 , - 1 < x < 1 , (5.1a) u ( - l ) = A _ i , u(l) = Ax . (5.1b) Here e > 0 is a small parameter, m > 0 is an integer and p(x) > 0 is an even smooth func-tion. This equation corresponds to the equilibrium problem of the exit problem studied in [79] and the references therein. One of the reasons to choose (5.1) as a model problem is that it is linear and its solution can be explicitly written in terms of a quadrature. This makes it easy to perform computations and comparisons. The second reason is that without a stability estimate, this equation still satisfies a comparison principle which is crucial for proving the convergence of a finite difference scheme. Despite its simplicity, we hope that the qualitative results revealed from this model equation are also applica-ble to other nonlinear metastable problems, for which a rigorous convergence analysis is typically not easy. Although there have been some error bounds for the finite element Galerkin method for the Cahn-Hilliard equation in [33] and [32], they do not guarantee any accuracy unless we use extremely small step size h . In fact, the constant C in error bounds Chr obtained there, where r is the convergence order, is dependent on the small parameter e and may be very large. Another loss of accuracy comes from the regularity assumption on the Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned EVP 127 solution u and its derivatives. However, the solutions u to the metastable problems are often associated with sharp boundary layers and/or internal layers where u will possess large derivatives in x as e —• 0. Therefore, in this section, we will give an analysis of uniform convergence introduced below. A discretization method is called uniformly convergent (with respect to e) of order r in the norm || • ||, if there exists a constant C that is independent of e and h, such that for all sufficiently small h (independent of e), || u-uh ||< ChT . (5.2) Here uh denotes a numerical solution obtained using this method. For illustration, we consider the singularly perturbed convection diffusion equation en" + u' = 0 , 0 < x < 1 ; u(0) = 0 , «(1) = 1 , (5.3) whose solution u — (1 — e~x^e)/(l — e -1 / ,< r) has a boundary layer at x = 0. The upwind scheme (see (5.24) below) on an equidistant mesh Ih = [xi = ih; i = 0 , 1 , . . . , N, Nh = 1} yields the numerical solution uh = (u,-) with m = (1 - /»•)(! - PNV , where p = (5.4) If the solution of (5.3) had bounded derivatives independent of e, a classical convergence theory for finite difference methods (i.e., consistency+stability =4> convergence) could guarantee the first order convergence of the upwind scheme. However, this convergence is not uniform in that (5.2) does not hold. In fact, when h — e, comparing ui = and u(xi) =  l~ e_\ , we can find that u(xi) — ui —> \ — e _ 1 ^ 0 as h —> 0, and l - e t consequently, the numerical solution uh does not converge to u(x) as h —> 0 uniformly. There are basically two approaches to construct uniformly convergent finite differ-ence schemes for the following singularly perturbed convection diffusion equation with Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 128 Dirichlet boundary conditions - e u x x + b(x)ux + d(x)u = f(x) , 0 < x < 1 , (5.5) under the assumptions that b(x) > b0 > 0 (i.e., no turning points), and d(x) > 0. One of the approaches is the exponentially fitting technique first introduced by Allen and Southwell [5] and Il'in [55], which yields the Il'in-type scheme. Through introducing an artificial diffusion by means of a fitting factor that enforces the scheme to be exactly satisfied by the boundary layer function in the leading order asymptotic expansion of the solution, the nodal convergence that is uniform with respect to the small parameter e was proved in [55] and [58]. Since discretization methods on equidistant meshes may have difficulties in representing the solutions that change abruptly in layers, an alternative approach to yield uniform convergence is the use of highly non-equidistant meshes (see [106], [107], [95] and [90] for details). One of the necessary steps in proving the uniform convergence of the numerical approaches above is to establish a uniform stability estimate for the discretization operator L h , such as \u{\ < C(max\LhUj\ + \u0\ + \UN\) , for i = 0 , 1 , . . . , N , (5.6) where C is a constant independent of e and h. Unfortunately, this type of uniform stabil-ity estimate obviously does not hold for any discretization of our model problem (5.1) due to its exponential ill-conditioning. To our knowledge, no numerical analysis addressing the uniform convergence of any finite difference scheme or finite element method for any metastable problem has been performed, even for the "simplest" model (5.1). In this chapter, we will study the uniform convergence of three finite difference schemes for (5.1): the upwind scheme, the coupled scheme and the Il'in scheme . This study is not only significant in understanding the finite difference schemes (and other nu-merical methods) applied to metastable problems, but is also interesting from the point Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 129 of numerical analysis: it provides an example showing that a uniform stability estimate is not a necessary condition for uniform convergence. The chapter is organized as follows. In the rest of this section, we estimate the derivatives of solutions it of (5.1) and we decompose u into a singular part and a less singular part. Using these analytical results, in §5.2, we construct an appropriate mesh generating function and three finite difference schemes and analyze the uniform convergence of these schemes on the corresponding meshes. Finally, in §5.3, we present some numerical results for (5.1) and discuss the numerical computation of some related exponentially ill-conditioned problems. 5.1.1 The Analytical Behavior of Solutions To study the uniform convergence of our difference approximations, it is necessary to investigate the analytical behavior of the solution to (5.1). The problem (5.1) is expo-nentially ill-conditioned, so it is obvious that the typical stability inequality || v ||< C || Lev || , for all v with v{-l) = v{l) = 0 does not hold, where C is a constant independent of £ . However, the comparison principle is still valid (cf. [82]), and its counterparts for difference schemes play an indispensable role in establishing uniform convergence. L e m m a 5.1 (Comparison principle) Suppose that v and w are functions in C 2 ( —1,1) f] C[0,1] that satisfy L£v(x) < L£w(x), for all x £ ( — 1,1) and v( — l) < w( — l), v(l) < w(l) . Then we have v(x) < w(x), for all x £ [0,1]. Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 130 Although (5.1) has a turning point at x = 0, nothing special happens near this point. For e —»• 0, its leading order boundary layer approximation has the form (cf. [79]) u(x) ~ u£(x) = \{A-i + A i ) + i(A_ a - A i ) ( e - ^ - e " ^ ) , (5.7) where £ = p ( ± l ) . To estimate the error in our discretization methods, we shall require bounds for the derivatives of the solution it to (5.1) that are valid for all small positive e . To analyze the Il'in scheme we need more precise information on the behavior of the solution. These results are contained in Lemmas 5.2 and 5.3. L e m m a 5.2 Let u(x) be the solution of (5.1). Then we have \u(x)\<max{\A_1\,\A1\} (5.8) and | u{l)(x) |< M£~' ( e - 7 ^ + e - 7 ^ ) , for i = 1,2,.... (5.9) Here 7 is any constant satisfying 0 < 7 < "^+2^ and the constant M > 0 is independent of £ . Proof. The proof of (5.8) is obvious by applying Lemma 5.1 with v = ±u and the barrier function w = max{| A _ i |, | Ai |}. Integrating (5.1) twice yields 1 1 e ^ ^ d s «(x) = 5M, + *_,) + - ^  e,-.M,)d3. where p(s) = /0S t2m+1p(t)dt. We can show that for x 2 m + 2 > e , r ee~^ds ~ 0 ^ , e^ 1 ^) as £ ^ 0. (5.11) Since p(x) — p(l) = — ^ f f 2 ( l — . T 2 m + 2 ) , where s0 6 (x, 1), the derivate of u satisfies u\x) = i ( A x - A_1)es_1^a;V /' e ^ ^ d s < M ^ e ^ ^ ' ^ 2 Jo < M e _ 1 e 2 - + 2 £ ( 1 * (5.12) Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 131 If we assume 7 is a constant satisfying 0 < 7 < , then (5.9) follows with i = 1. For i > 1, the result is obtained by induction and repeated differentiation of (5.1). • Lemma 5.3 Let u(x) be the solution of (5.1). Then it can be decomposed into u(x) = u e(x) + v(x), where uc(x) is given in (5.7) and v(x) satisfies I v {%)(x) |< M e 1 - ' ( e ~ 7 ^ + e - 7 ^ ) , for 1 = 0 ,1 , . . . ,4 . (5.13) Here the constant M > 0 is independent of £ and 7 is any constant satisfying 0 < 7 < m'mp(x) 2m+2 ' Proof. Since (5.1) is symmetric about x = 0, we prove (5.13) on 0 < x < 1 only. By symmetry, we have u(0) = | ( A _ i + Ai). So from (5.1) and (5.7), we find that v = u — ue satisfies Lev = -evxx + p(l)vx = / , 0 < x < 1, (5.14a) v(Q) = v(l) = 0, (5.14b) where / is defined by / EE (p(l) - X^p{x)) « ' - [A, - A_0 (p(l)f e - V * 1 ) 1 * 2 . (5.15) Given constants k, c and C\ satisfying k > 0 and c > c\ > 0, we have tke~ct < Ce~Clt for all t > 0, where C is a constant. Thus, using Lemma 5.2, we can show that \ f {t\x)\<M£- ie-^ , for i = 0 ,1 , . . . ,4 , (5.16) where 7 is any constant satisfying 0 < 7 < m 2 ' ^ ^ . Since L £ (exp (—7£ _ 1(1 — x))) = 7 (p(l) — 7) £~ l exp (—7£ _ 1(1 — x)), we may choose M large enough so that 4>(x) = Mee^ e~ l{l- x) ±v Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 132 satisfies Le(f> > 0, 0(0) > 0, 0(1) > 0. From the comparison principle, the inequality (5.13) holds for i = 0. We now integrate (5.14) twice and obtain where v. p (x) = - j z(t)dt , z(x) = /VvWe-8"1 J X J X (5.18) From (5.9) and (5.15), z(x) \< Me —ire x) (5.19) Hence | vp(x) \< Me. The constants Kx and K2 must satisfy Ki = 0 , v, Since /o1 exp(-e 1 p( l ) ( l - t))dt > Me, we have | K2 \< M. Therefore, from (5.17) and (5.19), we find that (5.13) holds with i = 1. The proof of (5.13) for i > 1 follows by Note that throughout this chapter we use M to denote a generic positive constant independent of e and the mesh width h and it may take different values in different formulas. Some of the constants will also be represented by M0 , Mi , C , Co , c, c 0 , etc. 5.2 Difference Schemes and their U n i f o r m Convergence 5.2.1 Difference Schemes and Some Prel iminar ies Denote the mesh Ih by Ih = {x{ : — 1 = XN < a-'-iv+i < • • • < XN-I <  XN — 1} with mesh widths hi = X{ — Xi„i and h = max; hi. Two meshes will be used in our discussion: (1). an equidistant mesh l£ = {xi = ih : i = — N,..., N, Nh = 1} ; and induction and repeated differentiation of (5.14). • Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 133 (2). a non-equidistant mesh = {xt : i = - T V , . . . , TV} , where with h = 1/N and ti = ih , ' A(tt + 1 ) - 1 , z = - / V , . . . , 0 , (5.20) l - A ( l - < , - ) , i = l , . . . , N . Here the mesh generating function A(i) (see Figure 5.1) is defined by \ ri>{t):=-ae\n(l-t/q), <G[0,a] , A(r) = < (5.21) _ 7r(t) := V>(a) + - a) , i € (a, 1] , where 4 € (0,1) and a > 0 are constants, and a G (0,g) is determined uniquely by 7r(l) = 1. Furthermore, some computations yield q-a = Me, ip(a) = -ae ln(l - - ) = -Me ln e, (5.22a) 1 < if>\a) < (1 - q)-\ 0 < \'{t) < </>'(a), 4>"(a) = Me~1. (5.22b) Using these estimates, we can easily show (cf. [104]) that I hi - hi-i \< M e ' x h 2 , for i = -N + 2 , . . . , N. (5.23) This non-equidistant mesh 1% is called Bakhvalov mesh [12]. If there were an ex-ponentially boundary layer at x = 0, then the boundary layer function would be y = exp(—0x/e), for some fixed 0 > 0. Bakhvalov's idea is to use an equidistant ?/-grid near y = 1 (which corresponds to x = 0), then to map this grid back to the x-axis by means of the boundary layer function. That is, grid points near x = 0 are defined by exp( — Pf-) — 1 — ^  , which is equivalent to Xi = —aeln(l — ^ ) with a = 0~l. The definition (5.21) produces a condensed grid near x = — 1 and x = 1, an equidistant grid outside the boundary layers and a gradual transition from the fine to the coarse grids. Using the usual notations for divided differences, D+Ui = ( u i + 1 - Ui)/hi+i , D-Ui = (m - Ui-i)/hi , D0Ui = (ui+i - Ui-i)l(hi + hi+i) , D+D-Ui = 2(D+Ui - Z>_u;)/(/\; + h i + 1 ) , Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 134 Figure 5.1: The mesh generating function A(t) versus t with a = 2 , q = 0.5 and e = 0.03. In this case, the tangent point of the curve ip(t) with the straight line ir(t) is a = 0.4618 . . . we will consider the following three finite difference schemes. (1). Upwind scheme on a non-equidistant mesh 1% , LhxUi = -eD+D-Ui + PlD'ul = 0 , -N <i< N , u_N = A _ x , uN = Aj , (5.24) where Pt = P(xt) = x2m+1p(xi), and D'Ui D + U i , if < 0 , D. if Pl > 0 . (2). Coupled scheme on a non-equidistant mesh Ih L2Ui — Lhcut = 0 , if P i < 1 , LhaUi = 0 , if P i > 1 , N < i < TV, (5.25a) (5.25b) Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 135 where with x{_i — | (x ,_ i + xA and P{_! = P(xi_i), f hi+1Pi/2e , if Pl > 0 , Pi = \ { -hiPi/2e , if P% < 0 , L^'ui = - t D + D _ u , + PiD0Ui , (central scheme [90]), (5.26) L^Ui = —eD+D-Ui + Pl±iD±Ui , (Gushchin-Shchennikow scheme [47]). (5.27) Here " ± " corresponds to whether Pi is negative or positive. (3). Il'in scheme on an equidistant mesh l£ , L\ui = -eaiD+D-Ui + PlD0ui = 0 , -N < i < N , u_iv = A-i , UJV = ^ i , (5-28) where the fitting factor <7; is defined by Oi = ^ coth ^ . The basic idea in constructing the Il'in scheme (5.28) is to select a fitting factor <r,- such that (5.28) would be accurate if the coefficient function P(x) were a constant P(xi). For the upwind and coupled schemes, we use a non-equidistant mesh which is dense near the boundary layers at x = ± 1 . For such a mesh, we can show that | u'(x) • x'(t) |< M. The coupled scheme is made up of a central scheme and a Gushchin-Shchennikow (G-S) scheme. Although the central scheme is usually unstable, the i-th. row of its corresponding coefficient matrix is diagonally dominant and has non-positive off-diagonal entries when pi < 1. On the other hand, in spite of the first order consistency of the G-S scheme, it becomes second order consistent at x = Xi if pi > 1. So the coupled scheme (5.25) is in fact stable and has second order consistency. The details of the background concerning these difference schemes and meshes can be found in [90], [98], [104], [106], [12], [55], [58] and the references therein. Here we give an explanation of why the schemes (l)-(3) are expected to be uniformly convergent on the corresponding meshes. First, for the singular perturbation problem (5.5) without turning points, the uniform convergence of these schemes has been proved in Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 136 [55], [58], [98], [104], etc. We wish to extend the analysis of uniform convergence in these studies to the exponentially ill-conditioned problem (5.1). Although (5.1) is a turning point resonance problem, the singularity of its solution only occurs in the boundary layers, which is similar to that of (5.5). Near the turning point x = 0, the solution of (5.1) is constant to within exponentially small terms and the derivatives are exponentially small for £ <C 1. In addition, although the (numerical) solution apparently does not satisfy the (discrete) stability estimate for (5.1), the (discrete) comparison principle is still valid for the turning point problem. Therefore, we anticipate that schemes (l)-(3) are uniformly convergent for (5.5) as well as for (5.1). To provide a rigorous analysis of the convergence of the difference schemes (l)-(3), we require some elementary facts about the difference operators L\, k — 1,2,3 . L e m m a 5.4 (Discrete Comparison Principle) For k = 1,2,3, the scheme L\u{ = 0, — N < i < N, with U _ J V = A _ i and u^ = A\, has a unique solution. If L\ui < L\vi, —N < i < N, and ifu-N < v-/v, UN <  V N , then Ui < u,- for —N < i < N . Proof. It is easy to show that the coefficient matrix of the scheme L\\ii =0, —N< i < N , with ii_jv and UN specified, is diagonally dominant and has non-positive off-diagonal entries. In addition, its rows corresponding to i = — N + 1 and % — N — 1 are strictly diagonally dominant. Hence, this matrix is an irreducible M-matrix and so has a positive inverse. Therefore, the solution Uj, —N < i < N, exists and if the u; are as specified in the lemma, then we have it,- < u,- for — Af < i < N. • From Lemmas 5.1 and 5.4 and the symmetry of the continuous problem (5.1) and the difference schemes (l)-(3), we get uo = u{0) = Ao = ^{A-i + A1). (5.29) Thus the proof of the convergence of the difference schemes (l)-(3) can be simplified to Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 137 studying the convergence of the following schemes Lhkut = 0, i = l,...,N-l, u0 = A0,uN = A1, k = 1,2,3, (5.30) for the continuous problem (5.1) on the interval 0 < x < 1. In a similar way as in establishing Lemma 5.4, we can show the following lemma. L e m m a 5.5 For k = 1,2,3, the scheme L\ui = 0, 0 < i < N, with Uo = A0 and UN = A\, has a unique solution. If L\uj < L\v{, 0 < i < N , and if u0 < v0, UN < vry, then U{ < Vi for 0 < i < N . The next lemma, with Lemma 5.5, will enable us to convert bounds for the truncation error into bounds for the discretization error. L e m m a 5.6 There exist positive constants /3, C\ and C2 depending only on x2m+1p(x) such that, for k — 1,2,3, Lhkrktl > { -C2e l , for i < N0, where N0 = mm{i : x, > |} and I and rk>i with i = 0,1, . . . , TV are defined by (5.31) N e ' = 1, rkii= TT TT~RT > fork = 1,2, (5.32) and I = 2, rKi = r l~N with r = e< , for k = 3. (5.33) Proof. A computation shows that If we select f3 satisfying 0 < <3 < § mini < x < 1 P(x), then for i = N0,..., N - 1, we h ave rh ^ M Ci £ + phi max(e, hi) Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 138 For i = 1 , . . . , N0 - 1, since T,?=i+i h3 > \ and £ f = ! + 1 h) < cN'1 , we have f.? = n a + ^ )>C E ^ = 5 ( ( E ^ 2 - E ^ M > — . - e2 So | L$ru \< M{e + Ph)- 1^ < C2e and we obtain (5.31) for i < N0. Next, since Lhcr2ii = Lhxru + Aru > l\ru , e + phi V  2 hi + hi+1 J where A = PiP 2hihi+\j'e(e + phi)(hi + hi+i) > 0, a similar argument can be used to obtain (5.31) when k — 2. For k = 3, a computation gives Lh3r3>l = ^-(r-l) 2Ar3tl, (5.35) where A = coth = coth coth - smh / smh — smh . r - 1 2e 2e 2e 2e 1 2e 2e For i > N0 , (5.31) can be proved in the same way as Lemma 4.2 in [58]. For i < No, since c\t < s inhi < c2t for 0 < t < c, we have | A |< |r f if /i < e. In this case, using r~ x (r - l ) 2 < Me~ 2h 2 and r3,< < Me3, we obtain (5.31) from (5.35). If e < h, then since (r + 1) < c(r — 1) and tcotht < c(l + £) for t > 0, from (5.35), we have Mpr p.]i I ^ 3 , , | < — ^ - coth -^-r3l< < Me~ lr3^x < C2e2 . The proof is completed. • Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 139 5.2.2 Convergence Analys i s In this section, we derive error bounds for our difference schemes (l)-(3). Let u(x) be the solution of (5.1) and u\ the solution of the system L\u\ = 0, 1 < i < N — 1, with UQ = A 0 and uN = A x . Our first result is T h e o r e m 5.1 Let 7 be given in Lemma 5.2 and the non-equidistant mesh 1% be de-fined by (5.20) with 07 > 4 . Then the coupled scheme on 1% is second order uniformly convergent, i.e., \u{xi) -u 2 |< Mh 2 , z = 0 , . . . , / V . (5.36) Proof. We first consider the truncation error associated with the coupled scheme. A computation in [104] has showed that r 2 EE / ^ ( ^ ( X ; ) — u 2) satisfies \T2\<M ((hi+1 - hi)e- 2Ve{xi) + h 2e- 3VE(xl+1)) , (5.37) where V£(x) EE exp(—7(1 — x)/e). Lemma 5.2 gives that the solution u(x) to (5.1) on [0,1] has the same differentiability properties as the problem (5.5), which does not have any turning points. So the same argument as that in [104] can be used to derive a bound for T 2 as follows I r 2 \< C3h2 (l +  17-T\ r^) , i = 1, . . . , JV - 1, (5.38) \ max(e, «,•) J if 07 > 4. Furthermore, this bound can be improved when i < N0. Since e~3V£(x) < Me-il' for x < \ , with some 0 < 7 < 7/2, from (5.23) and (5.37), we have I r 2 |< C 4 / i 2 e - 7 / £ , for i < No . (5.39) Thus to estimate the discretization error e\ = u(xi) — u 2, we define 10; = M\h 2<f)i + M2h 2r2,i ± e\ , where fa = Axi — x 2. Since x 4 _ i > ^x; for i > 1, it is clear that L\fa > mm{Lhcfa, Lhafa} > C0(e + x 2 m + 1 ) , (5.40) Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 140 where L% = 2s + i*(4 - x^ - xt+1) and L% = 2e + P 2_i (4 - x{ - x^). Using (5.38), (5.39), (5.40) and Lemma 5.6, we have . . f MxCQ{e + x2m+1) + M2 ? , ,r2-C3(l + — h n ^ i) , * > , jl^l^yj. > J zmax(e,h.;) ^' ° V max(E,/i() *>V ' — u ' ' ~ } M i C 0 ( e + x2m+1) - M2C2e - C4h2e~^ s, i < N0 . (5.41) Noting that C o , . . . , C 4 are fixed constants, we may first choose M2 = M2(C\, C3) large enough such that the sum of the coefficients of r 2 i ; in (5.41) is positive. Then we choose Mi = M\(M2, Co, C2, Czi CA) large enough such that the right hand sides of (5.41) are positive. In addition, it is obvious that WQ > 0 and > 0. Thus, Lemma 5.5 yields iol > 0, i.e., | e\ |< Mxh 2fa + M2h 2r2,i < Mh 2 , for i = 0 , . . . , JV . (5.42) • Similarly, we can prove, for the operator L\ , Theorem 5.2 Let 7 be given in Lemma 5.2 and the non-equidistant mesh l£ be defined by (5.20) with cry > 2 . Then the upwind scheme on I\\ is first order uniformly convergent, i.e., \u(xi) -u) |< Mh , i = 0 , . . . , i V . (5.43) We now give an error bound for the Il'in scheme. T h e o r e m 5.3 The solution u 3 to the Il'in scheme on the equidistant mesh l£ satisfies h 2 \u{xi) -u 3 \< M-j—^—, i = 0,...,N. (5.44) Proof. Again, since the solutions to (5.1) on [0,1] and (5.5) have the same differentiability properties and both can be decomposed into a singular part £ exp(— p ( l ) e _ 1 ( l — x)) (( is some constant) and a less singular part as in Lemma 5.3, we can use the reasoning in Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 141 [58] to obtain the following bound for the truncation error r 3 = L^u(xi) - u3) of the Il'in scheme I rf |< C3(e + h)- lh2 ( l + e-Vg,,-) , i = l,...,N-l. (5.45) For i < No , this estimate can be refined. In fact, since | <rt- — 1 |< Me~ 2h 2, using a Taylor expansion, it is easy to show that if Xi < |, I rf \< Mh 2e- 3V£(xi+1) < C^e^1* , with some 0 < 7 < \p , (5.46) where K(^ ) = exp(—7(1 — x)/e). Let 0(rc) = 4x — x 2 . Then, noting that <7t- > 1, we have L% = 2eal + P,(4 - 2a;,-) > C0(e + x2m+1), i = 1, . . . , TV - 1. (5.47) If h < e , then we define Wi = e" 1 / i 2 (M 1 0 , + M 2 r 3 i t ) ± (u(a;2) - u 3 ) . Using (5.45), (5.46), (5.47) and Lemma 5.6, we obtain \ M aCo(£ + x 2 m + 1 ) + M 2 C 7 l £ - V 3 , 4 - C 7 3 ( l + e - 1 r 3 , J ) , i > N0, , ^ eh L%Wi > < (5.48) [ MxCo(e + x2m+1) - M2C2e 2 - C4ee~^ s, i < N0 . Since we can choose suitable positive constants M a and M2 such that the right hand sides of (5.48) are positive, Lemma 5.5 yields w{ > 0, i.e., I u(xi) - u3 \< e~1h2(M1(j)i + M2r3ti) < Me~ xh 2, if /> < e . (5.49) If e < h, let ivi = h(Mi4>i + M 2 e _ 1 ^ r 3 i i ) ± (u(a:,-) — u 3 ) , which satisfies ft-1!V > / M l C ° { £ + ^ 2 m + 1 ) + M 2 £ _ 1 C l ^ - <?3(1 + £-^3,0 , i > No , . 3 ^ (5.50) M i C 0 ( e + . r 2 , n + 1 ) - M2ehC2 - C4he'^ s, i < TV0 . Again, there exist constants Mi > 0 and M 2 > 0 such that L3Wi > 0. So, since w0 > 0, wN > 0 and e _ 1 / *r 3 ) i < £ - 1 / i exp(-/?£ _ 1 / j ) < M for i < N - 1, Lemma 5.5 gives I u{xi) - u3 \< h ( M i 0 i + M2e~ lhrztf) < Mh , if e < fc. (5.51) This completes the proof of the theorem. • Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 142 5.3 Numerical Experiments and Discussions The convergence analysis in the previous section indicates that the exponential i l l -conditioning of the singularly perturbed resonance problem (5.1) does not cause any special difficulties in its numerical computation different from the non-turning point problem (5.5). Thus the discretization techniques used to construct uniform convergence schemes for (5.5) can lead to uniform difference approximations for (5.1) as well. We now present a few numerical results to examine our analysis. 5.3.1 A Model Problem of (5.1) To verify the uniform convergence of our schemes (5.24), (5.25) and (5.28) numerically, we consider a specific problem of the form (5.1) -eu" + xu' = 0, - K x < l , u ( - l ) = - 3 , u(l) = 1. (5.52) We solve this problem for various values of e and h. In Tables 5.1, 5.2 and 5.3, we list the maximum error E^ and the numerical convergence order r of our schemes L\, = 1,2,3 respectively, for e — 0.01, 0.02 and 0.03, where E^ = max, | u(x,) — u1- \ and r = (log E^ — log E^[2)j log 2 . Here u(x) is the analytical solution of (5.52) and v!-the solution of one of our schemes with mesh size h = 1/N. For A(r) defined in (5.21), we use 9 = | and a = 2. The computation here was performed in F O R T R A N double precision (approximately 15.95 significant decimal digits) on a HP735 at U B C . The solver for computing the linear systems corresponding to our schemes is Gaussian elimination and the exact solution u(x) is obtained by applying a numerical quadrature with suffi-cient precision to (5.10). Note that the principal eigenvalue of the eigenvalue problem associated with (5.52) satisfies A 0 » 2.662 x 10 - 7 for e = 0.03, A 0 « 7.835 x 1 0 - 1 1 for e = 0.02 and A 0 « 1.539 x 10 ' 2 1 for e = 0.01 (cf. [79]). Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned EVP 143 Let's analyze Table 5.2 first. Table 5.2 shows that the coupled scheme is uniformly second order convergent only when the number N of mesh points is not too large. For large N, on the contrary, the convergence of the scheme deteriorates, which contradicts the classical convergence theory that the discretization error of a convergent scheme will tend to zero as the mesh size goes to zero. In addition, the smaller e is, the sooner this deterioration happens as iV increases. Since the uniform convergence of the coupled scheme has been established analytically in §5.2, the only explanation of this paradox is that the Gaussian elimination does not give an accurate numerical approximation to the difference scheme because of the round-off errors of a computer. Specifically, we believe that our schemes do converge uniformly in e as h —> 0, and the degeneration of the discretization errors for large N is caused by the round-off errors in computation due to the severe ill-conditioning of the finite difference operators. We partially verify this explanation by calculating and presenting the condition numbers of the coefficient matrices of our schemes in Table 5.4 for different values of e and h. Note that since these condition numbers are calculated by M A T L A B which uses double precision, they may be NOT accurate especially when they are greater than about 10 1 5 . Comparing Table 5.4 with Table 5.2, we notice that the numerical convergence order r for the coupled scheme, roughly speaking, begins to deviate from the analytical value of 2 when the condition number is close to 10 1 5, which is approximately the reciprocal of the (double) machine precision. Thus, we believe that this degeneration of the convergence can be avoided if we can calculate a difference scheme more precisely. Based on this explanation, it is anticipated that the numerical computations with quadruple precision arithmetic will yield better results than those with double precision. This is verified in Table 5.5, where the second order convergence for the coupled scheme is obviously obtained using quadruple precision arithmetic even for much larger N than those in Table.5.2 and e = 0.01. On the other hand, we expect that single precision arithmetic will not give good Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 144 numerical results in solving our difference schemes. This is also verified numerically in Table 5.6, where we get only very few significant digits of accuracy even for £ = 0.03 and when h is large. From Tables 5.1 to 5.6, we can notice that our previous observations about the coupled scheme are also applicable to the upwind scheme and the Il'in scheme. Here we give a further discussion of these schemes. It is clear for each scheme that for a fixed h, the condition of a difference scheme becomes worse if e is smaller. Thus, for a larger value of e , it is natural that we can solve our schemes accurately over a wider range of values of h . For fixed values of e and h , we notice that the condition number of the coupled scheme is greater than that of the upwind scheme. That means a higher order scheme can only be solved accurately for a larger h. However, since a higher order scheme with a larger h may give a better result than a lower order scheme with a smaller h, we find from Tables 5.1 and 5.2 that the coupled scheme resolves the exponentially ill-conditioning problem better than does the upwind scheme. In summary, although the discrete stability estimate is not valid for a numerical scheme of an exponentially ill-conditioned problem, a truncation error rh may not result in very large errors in the numerical solution uh and a scheme may still be uniformly convergent with respect to e. However, a numerical method will usually inherit the ill-conditioning associated with the continuous problem. This causes the peculiar phe-nomenon we observed in the computations that a small number of meshpoints N may give better numerical results than does a large value of N. To minimize the effects of round-off errors that may pollute the accuracy of a scheme, higher (such as quadruple) precision arithmetic is preferred in solving the discrete linear systems. Furthermore, our numerical experiments suggest that higher order difference schemes (or other numerical methods) are usually more efficient in computing the numerical solutions of exponentially ill-conditioning problems. Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 145 N e = 0.03 £ = 0.02 £ = 0.01 Eh oo r Eh CO r Eh CO r 24 .1491e0 0.93 .1532e0 0.93 .1585e0 0.95 48 .7840e-l 0.97 .8020e-l 0.97 .8211e-l -1.21 96 .4010e-l 0.99 .4086e-l 0.99 .1897e0 -192 .2026e-l 0.99 .2060e-l 0.99 - _ 384 .1018e-l 1.00 .1039e-l 0.89 -768 .5105e-2 1.00 .5616e-2 0.80 - _ 1536 .2556e-2 1.00 .3227e-2 -3.29 3072 .1282e-2 - .3150e-l - - -Table 5.1: Numerical results of the upwind scheme for (5.52) using double precision. 5.3.2 A Nonlinear Problem and a Time-dependent Problem Our purpose in carrying out the next two numerical experiments is to illustrate that our preceding analysis regarding the numerical computation of the model problem (5.1) might also be applicable to other types of exponentially ill-conditioned boundary value problems and their corresponding time-dependent equations. The first test problem is the steady state Ginzburg-Landau equation £2uxx + Q{u) = 0, - K x < l , ux(-l) = ux(l) = 0 , (5.53) with Q(u) = 2(u — u3). For this nonlinear B V P , one of its solutions that can be asymp-totically approximated by u ~ ue(x) = tanh(x/e) has one internal layer at x = 0. So, to compute this solution, we use a non-equidistant mesh 1% = {x{ : i = —N,... ,N}, which is dense near x — 0 and equidistant outside the internal layer, where with the same notations as in (5.20), it\ \ x { t l h * = o,...,tf, X i - x(ti) = < (5.54) [ - A H , ) , i = - N , . . . , - l . Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 146 N e = 0.03 e = 0.02 e = 0.01 r r El r 12 .2545e-l 2.41 .2184e-l 1.89 .1900e-l 2.04 24 .4789e-2 2.00 .5872e-2 2.15 .4615e-2 -7.77 48 .1199e-2 1.99 .1319e-2 1.87 1.01 _ 96 .3017e-3 1.98 .3598e-3 0.22 - _ 192 .7659e-4 1.99 .3088e-3 -0.25 -384 .1925e-4 1.79 .1466e-2 -1.10 - _ 768 .5558e-5 -0.76 .3136e-2 -0.97 - _ 1536 .9406e-5 - .6162e-2 - - -Table 5.2: Numerical results of the coupled scheme for (5.52) using double precision. Here X(t) is defined by X(t) = (5.55) ip(t):=-aet/(q-t), te[0,a], { ir(t) := V(a) + ^'(a)(t -a), t e (a, 1] , where q = \ , a = 1.5, and a £ (0, q) is determined uniquely by 7r(l) = 1. This mesh is similar to (5.20) except that o can be calculated explicitly now. On this mesh, we apply the following finite difference scheme e 2D+D_Ui + Q(Ui) = 0, -N < i < N, D+u_N = D_uN = 0 (5.56) to solve (5.53). Since the exact solution of (5.53) we are seeking is constant near the end points x = ± 1 to within exponentially small terms, first order precision in the difference approximations of the boundary conditions will not ruin the second order accuracy of the difference operator on the interior points when e is small. In addition, it is clear that | d*u£/dx* \< M e - ' exp(—^\x\/e) with some constant 7 > 0, which is similar to the differential properties of the solution of (5.1). So we expect that the scheme (5.56) on the Bakhvalov type mesh (5.54) is second order uniformly convergent with respect to e as h —> 0. In our computations, the nonlinear discrete system corresponding to Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 147 N e = 0.03 e = 0.02 £ = 0.01 Eh CO r Eh CO r oo r 12 .7158e-2 1.06 .3890e-2 0.05 O.lOOel 0.00 24 .3424e-2 1.97 .3766e-2 1.49 O.lOOel -48 .8765e-3 1.93 .1337e-2 1.88 - -96 .2296e-3 2.00 .3633e-3 1.99 - -192 .5755e-4 1.99 .9157e-4 -1.33 - -384 .1447e-4 1.87 .2309e-3 -4.08 - -768 .3945e-5 0.90 .3899e-2 -0.94 - -1536 .2116e-5 - .7459e-2 - - -Table 5.3: Numerical results of the Il'in scheme for (5.52) using double precision. N Upwind scheme (5.24) Coupled scheme (5.25) Il'in scheme (5.28) £ = .03 £ = .02 £ = .01 £ = .03 £ = .02 £ = .01 £ = .03 £ = .02 £ = .01 12 4.17e6 5.64e7 6.39e9 8.67e7 4.87e9 7.31e9 1.14e8 2.66ell 1.81el7 24 1.75e8 7.44e9 1.07el3 2.30ell 3.20el3 2.60el4 4.74e8 1.21el2 8.00el6 48 4.94e9 7.98ell 3.46el5 1.64ell 6.31el4 4.04el7 1.89e9 4.77el2 3.07el6 96 8.02el0 4.50el3 1.52el9 5.38ell 4.48el5 1.33el7 7.56e9 1.89el3 4.86el6 192 7.79ell 1.13el5 1.40el7 2.15el2 1.16el6 2.12el7 3.02el0 7.52el3 6.41el6 384 5.20el2 1.46el6 - 8.80el2 5.00el6 - 1.21ell 3.03el4 -Table 5.4: Numerical results (vising Matlab) of the condition numbers || A \\2 H A 1 ||2 of the coefficient matrices A of schemes (5.24), (5.25) and (5.28). (5.56) is solved by Newton's method with u e(x) as the initial guess. The iterations are stopped when the maximum pointwise absolute difference between successive iterations becomes smaller than en, which equals 1 0 - 1 ° for double precision arithmetic and l O - 2 0 for quadruple precision arithmetic. The numerical results are compared to the numerical solution obtained by the same method with a sufficiently large N = No- The corresponding maximum pointwise abso-lute errors E^ , as well as the numerical rate r\ and anticipated rate r2, are presented in Tables 5.7 and 5.8, for e = 0.05, 0.1 and 0.15, where r x = EhJE^2 and r2 is defined Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 148 N 48 96 192 384 768 1536 Upwind scheme (5.24) Er C O .8204e-l .4153e-l .2084e-l .1044e-l .5222e-2 .2611e-2 0.98 0.99 1.00 1.00 1.00 Coupled scheme (5.25) E: .1225e-2 .3310e-3 .8265e-4 .2064e-4 .5154e-5 .1286e-5 1.89 2.00 2.00 2.00 2.00 ITin scheme (5.28) .1815e-2 .6603e-3 .1675e-3 .4221e-4 .1075e-4 .2875e-5 1.45 1.97 1.99 1.97 1.90 Table 5.5: Numerical results of schemes (5.24), (5.25) and (5.28) using quadruple p arithmetic. Here e = 0.01. recision N Upwind scheme (5.24) Coupled scheme (5.25) I l ' in scheme (5.28) r Et r El r 12 24 48 96 .2699 .1540 .0998 .4016 0.81 0.63 -2.01 .0289 4.492 .2234 1.113 -7.28 4.33 -2.32 .3584 .4987 .6670 .8627 -0.48 -0.42 -0.37 Table 5.6: Numerical results of schemes (5.24), (5.25) and (5.28) using single precision arithmetic. Here e = 0.03 . below. Note that, from (3.23), the principal eigenvalue of the eigenvalue problem associ-ated wi th the linearized equation about the one layer solution satisfies A 0 « 2.518 x IO - 1 0 for e = 0.15, A 0 « 4.078 x IO" 1 6 for e = 0.1 and A 0 » 1.733 x 10~ 3 3 for e = 0.05. Let us estimate what the values of r 2 should be. Let h0 = 1/NQ and suppose h = 2kh0 wi th some integer k > 1. If our scheme is second order uniformly convergent, then we have, by assuming that uh(xi) ~ u(xA + c(xi)h2 , un - uh0 - c(2 lk - l)h, So we set the anticipated rate uh'2-uh° - c ^ " 1 ) -l)h2Q. r2 = 4k - 1 4 * - 1 - 1 (5.57) Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 149 From Table 5.7, where double precision arithmetic is used, we notice that the values of ri and r 2 agree with each other when the number N of the mesh points is not large. However, it is true for all values of £ listed that the numerical rate r\ begins to diverge from the anticipate rate r 2 when N is greater than about 1024 and Newton's method does not converge when TV > 1024. Although we can't explain exactly what causes this value of 1024, which should depend on the value of e, to be independent of e , from our experience in computing the solution to (5.1), we speculate that the disagreement in Table 5.7 stems from the round-off errors in computation. This is verified by Table 5.8, where we use quadruple precision arithmetic. Surprisingly, in Table 5.8, r\ agrees with r2 to more than five significant digits when N > 2048. e = 0.15 £ = 0.1 £ = 0.05 N : k r2 E h C O Eh C O r\ El n 16:9 4.0000 .4110e-4 3.8297 .1094e-3 4.0286 .1478e-3 4.0767 32:8 4.0002 .1073e-4 4.0662 .2715e-4 4.0008 .3625e-4 3.9745 64:7 4.0007 .2639e-5 3.9700 .6785e-5 3.9999 .9120e-5 4.0044 128:6 4.0029 .6648e-6 4.0087 .1696e-5 4.0024 .2277e-5 3.9984 256:5 4.0118 .1658e-6 4.0131 .4238e-6 4.0116 .5696e-6 4.0117 512:4 4.0476 .4132e-7 4.0177 .1057e-6 4.0463 .1420e-6 4.0457 1024:3 4.2000 .1029e-7~ 3.5997 .2611e-7~ 3.9447 .3509e-7~ 3.8957 2048:2 5.0000 .2857e-8~ 0.5854 .6619e-8~ 1.5100 .9008e-8~ 0.8692 4096:1 - .4881e-8~ - .4384e-8~ - .1036e-7~ -Table 5.7: Numerical results of the scheme (5.56) using double precision. Here Ao = 8192 , £ n = 1 0 - 1 0 . For those E^ ending with " ~ " , Newton's method does not converge within 20 iterations. For our second test problem, we consider the time-dependent problem corresponding to (5.1) from [79] ut = euxx — xux, — l < x < l , t>0, (5.58a) u(-l,t) = 2, « ( ! , * ) = u(x,0) = ^ + 3 ( x - l ) 2 / 8 . (5.58b) Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 150 e = 0.15 £ - 0.1 £ = 0.05 N : k r2 El OO ri Eh C O r\ El r\ 32:9 4.0000 .1073e-4 4.0660 .2715e-4 4.0007 .3625e-4 3.9744 64:8 4.0002 .2639e-5 3.9694 .6786e-5 3.9994 .9120e-5 4.0038 128:7 4.0007 .6649e-6 4.0065 .1697e-5 4.0002 .2278e-5 3.9962 256:6 4.0029 .1660e-6 4.0043 .4241e-6 4.0028 .5700e-6 4.0032 512:5 4.0118 .4144e-7 4.0121 .1060e-6 4.0117 .1424e-6 4.0118 1024:4 4.0476 .1033e-7 4.0477 .2641e-7 4.0475 .3549e-7 4.0476 2048:3 4.2000 .2552e-8 4.2000 .6526e-8 4.2000 .8769e-8 4.2000 4096:2 5.0000 .6076e-9 5.0000 .1554e-8 5.0000 .2088e-8 5.0000 8192:1 - .1215e-9 - .3107e-9 - .4176e-9 -Table 5.8: Numerical results of the scheme (5.56) using quadruple precision. Here iV 0 = 16384, £ n = I O " 2 0 . It is shown in [79] that the metastable dynamics for (5.58), valid away from an initial time layer, can be asymptotically described by u(x,t) ~ ue[x] A(x,e)] = A(t,e) + (2 - A{t,e))e-^ + - A(t,e)j e'1^ , (5.59) where with A 0 = \J~^(1 - £ + 0(e 2)) e~^i , A(t,£)~5--3-(l-£)e-^. (5.60) To compute u(x,t) numerically, we apply the T M O L in §1.3 to (5.58) except that the B V P solver COLSYS is replaced by our difference scheme (5.24), (5.25) or (5.28), where the mesh generating function X(t) is given in (5.55) instead of (5.21). From our numerical results, we output the value of u(0,t), which gives the numerical prediction for A ( t , t ) . In Figure 5.2, 5.3 and 5.4, we plot t versus A for the asymptotic result (5.60) (dotted lines) and the numerical results (solid lines) corresponding to the upwind scheme, the coupled scheme and the Il'in scheme, respectively, for various values of h. Here the computation was performed in double precision arithmetic. The figures indicate that two curves representing the asymptotic and numerical results of t(A) converge as h decreases. Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 151 For the center scheme with N = 400 and the Il'in scheme with N > 100, the two curves are virtually indistinguishable. These phenomena suggest that our numerical methods are convergent and can be used to give a numerical verification of the validity of the asymptotic metastable dynamics. Based on our convergence analysis and numerical experiments, we believe that classic finite difference schemes and other numerical methods may be applied to compute the singularly perturbed problems exhibiting dynamic metastability as long as the mesh size h is suitably selected and e is not too small. In such cases, the exponentially ill-conditioned singularly perturbed problems do not cause more troubles in numerical computations than other "standard" singular perturbation problems such as (5.5) do. c: N = 2000 d: N = 4000 Figure 5.2: Plot of t versus A for (5.58) from the asymptotic approximation (dotted line) and from the full numerical approximation (solid line) using the upwind scheme when £ = 0.025. Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 152 0.85 0.0 0.95 5 1.2 1,25 1.3 a: N = 100 b: N = 200 0.85 0.9 0.95 c: N = 300 d: A T = 400 Figure 5.3: Plot of t versus A for (5.58) from the asymptotic approximation (dotted line) and from the full numerical approximation (solid line) using the coupled scheme when e = 0.025. 5.3.3 A Spectral M e t h o d According to our observations in §5.3.1, we believe that higher order numerical methods, which can lead to small discretization errors with a moderate number of mesh points, are usually more effective to compute the numerical solutions of exponentially ill-conditioned problems. Since the spectral methods are known to be able to offer exponential accuracy meaning that the error between the numerical solution and exact solution decays expo-nentially versus N, we now introduce a new procedure based on coordinate stretching and the Chebyshev pseudo-spectral (PS) method to solve (5.1). Our aim is to see with this procedure if and then how significantly we can improve the numerical results in §5.3.1. Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 153 c: N = 200 d: N = 300 Figure 5.4: Plot of t versus A for (5.58) from the asymptotic approximation (dotted line) and from the full numerical approximation (solid line) using the Il'in scheme when e = 0.025. It has been found (see [25], [42]) that the PS method is attractive in solving singular perturbation problems having boundary layers by clustering the mesh points toward the boundaries, for example, as in the Chebyshev method (x{ = cos^,z = 0, . . . , i V ) . However, to obtain an accurate solution with a very small parameter e, a large TV is required to guarantee that at least one of the collocation points lies in the boundary layer. To avoid this difficulty, Tang and Trummer [105] introduced a sequence of SINE transformations of the computational domain so that some collocation points are within a distance e from the boundaries a; = ± 1 for e •C 1. Indeed these transformations together with the Chebyshev PS method can deal well with very small boundary layers with a fairly small number of collocation points. However, we notice that since the SINE transformation does not make use of the properties of a boundary layer such as (5.9), Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 154 it does not ensure that at least one collocation point lies in the boundary layer for all e > 0. Thus, with a fixed number of the SINE transformations, the number of collocation points required to resolve a boundary layer is still dependent on e. Instead of the SINE transformation, therefore, we consider the following coordinate transformation, which yields O(N) collocation points, independent of e, in both the boundary layers and the interior domain: f A(* +1) - i , te [-1,0], \ l - A ( l - f ) , t € ( 0 , l ] , x(t) = (5.6i; where \(t) = \ ^ : = ~Ae l n I1"2(1 ~ £ ^ t 0 < t < 1 2 ' (5.62) Here A > 0 is a constant, K > 0 is an integer and B = B(t, e) is determined such that x(t) is a strictly increasingly function connecting (—1, —1) and (1, 1) on the x-t plane and has certain required smoothness. Another advantage of the coordinate transformation (5.61) over the SINE transformation is that the solution in the computational coordinate t can be as smooth as needed by selecting the integer A'(see (5.65) below). This is rather important to the convergence of the spectral methods, since lack of smoothness of a continuous solution is the main source of degradation of the expected infinity-order accuracy for a spectral method due to the Gibbs phenomenon. The transformed form of (5.1) can be written as -ev"(t) + P(t)v'(t) = 0 -1< t < 1 , v(-l) = A _ a , v(l) = Ax (5.63) where v is the transplant of u, v{t) = u(x(t)), and P(t) = + x(t)2m+1p(x(t))x'(t). To ensure that this transformed equation is well defined, we require that x(t) has at least up to second order continuous derivatives and x'(t) > 0. It is clear that \{t) is K-th Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 155 order continuously differentiable and if we let TT(1) = 1, TT"(1) = 0, (5.64) then x(t) has up to third order continuous derivatives. In our computations, we let B = bo + bi(t — | ) , where bo and &i are determined by (5.64). Now let us show that for this choice of B, (5.63) is well defined and its solution v{t) has uniformly bounded derivatives up to iv-th order except at t = 0 when A-y > K(K + 1), i.e., | v{k\t) \<M for 0 < |t| < 1 and k = 0 , . . . , K , (5.65) where 7 is the constant given in (5.9) and M > 0 is a constant independent of e. First, to show (5.63) is well defined, it suffices to prove X'(t) > 0. From (5.62), we have 4>W(t) =  A ^ I j^f6 , ^ (k\\) = A(k - , (5.66) for k = 1, . . . , K, where q = 2(1 - £ * " T T ) . Since ^ k ) {\) 0 as e -»• 0, from (5.64), we obtain 60 ~ 2K(K + 2) , bx ~ -K2K+1 , as e -> 0 . (5.67) It follows that on | < t < 1, ^ + 6 1(/.+2)(f-i; Tr'(t) > 6d(A' + l ) ( t - - ) i r  1(A' + < i)/f+1 = (/i- + 2 ) 2 A " ( ^ - i ) A " ^ 4 - 2 - 2 / , ( t - i ) ) > 0 , a s £ ^ 0 . The proof of \'(t) > 0 on 0 < t < \ is apparent. Next, let's consider (5.65). By symmetry, we only show it on -1 < t < 0. For -1 < t < -\, using (5.9) and (5.66), we have , dv du dx _ 1 -> ie in( i - 9 ( t+i) ) A£o — = — • — < M.£ e e • dt dx dt l _ 1 - q(t + 1) < M(l - q(t + l))^-1 < M , i f A 7 > l Chapter 5. Numerical Analysis of an Exponentially Ill-Conditioned BVP 156 and similarly, for k = 1, . . . , K , | ^ |< M (1 - q{t + l ) ) - 4 ^ < M , if A 7 > k . For - | < t < 0, since A(t + 1) > ^(|) = - A g l n e / ^ + 1), we have 1^1 < M \ p ^ \ < M e - k e - ^ l x ^ dt" dxk < Me^~k < M , if A-f > k(K + 1) . Thus, (5.65) holds when A^ > K(K+l). It is certain that we can find a mesh generating function A(f) such that (5.65) is true on the whole region of t i.e., including t = 0 . But since the solution of (5.1) is constant to within exponentially small terms outside the boundary layers, we do not expect doing so will make any significant improvements. Our numerical procedure is to solve the transformed problem (5.63) using the standard Chebyshev PS method (cf. [25], [42]). The resulting linear system is solved by Gauss elimination in double precision. Our numerical experiments displayed in Table 5.9 show that this procedure is indeed superior to our finite difference schemes in §5.3.1 for (5.1). For example, for e — 0.01, the least maximum error we can obtain in Table 5.9 is about O(10 - 6 ) , whereas it is only at best O(10 - 2 ) in Tables 5.1 to 5.3. In addition, we can still obtain four significant digits of accuracy even for (5.52) with e — 0.002, for which the corresponding principal eigenvalue A 0 ~ O(10~ 1 0 8 ) . Chapter 5. Numerical Analysis of an Exponentially 111-Conditioned BVP 157 N £ = 0.03 £ = 0.02 £ = 0.01 £ = 0.005 £ = 0.002 10 .5635e-2 .7848e-2 .1753e-l .3859e-l .7900e-l 20 .8912e-5 .6488e-5 .6045e-5 .7707e-4 .1222e-3 40 .3107e-6 .6807e-6 .2759e-6 .1831e-5 .2806e-4 80 .2949e-7 .7160e-7 .2487e-6 .1997e-5 .2787e-4 120 .7312e-8 .5419e-7 .8624e-6 .1978e-5 .2790e-4 160 .8513e-8 .6267e-7 .2030e-3 .5149e-5 .2797e-4 200 .9894e-8 .7220e-7 .1724e-l .6075e-5 .2817e-4 240 .9094e-8 .4613e-7 .1303e0 .1725e-3 .14286-21 Table 5.9: Maximum errors for the spectral method applied to (5.52) using double pre-cision. Here, K = 3 and A = 8. ^ o r this entry, N=400 Chapter 6 Summary and Future Work 6 .1 Summary In this thesis we have applied asymptotic and numerical methods to investigate meta-stable behavior associated with several time-dependent singular perturbation problems, including the generalized Burgers equation modeling an upward flame front propagation in a vertical channel, the viscous Cahn-Hilliard equation modeling the phase separation of a binary mixture and two problems related to exponentially slowly varying geometries. Specifically, we employed the projection method to derive ordinary differential equations (ODEs) or differential algebraic equations (DAEs) for the undetermined constants in the conventional M M A E solutions to the equilibrium problems. From these ODEs/DAEs , the metastable behavior was then studied quantitatively and in detail, and the equilibrium solutions and their stability were also obtained. In addition, the principal eigenvalues of the linearized operators were estimated asymptotically for the flame front problem and the slowly varying geometry problems. Most of our crucial asymptotic results were verified by the full numerical results computed using the T M O L . Another role of our numerical method T M O L is to provide useful information about the metastable solutions in their transient phases and collapse phases during which our asymptotic analysis fails. For the flame front problem, it was suggested by Berestycki, et al. [16] that the parabolic flame front may be dynamically metastable in the sense that its tip location 158 Chapter 6. Summary and Future Work 159 remains near its initial location for an exponentially long time. However, to our knowl-edge, no rigorous proof has been given yet and little is known of its detailed dynamics. In this thesis, we considered a generalized form of the Burgers type equation. It was shown that the principal eigenvalue associated with the linearization around the equilibrium is exponentially small. In addition, the metastable behavior was studied quantitatively by deriving an asymptotic ODE characterizing the slow motion of the tip of a parabolic-shaped interface. Our asymptotic results were shown to compare very favorably with full numerical computations and give a first detailed and quantitative description of the metastable flame-front motion in a vertical channel. For the Burgers-like convection-diffusion equation which describes one dimensional transonic flow through a nozzle with a slowly varying cross-sectional area, we studied the effect of an exponentially small change in the cross-sectional area upon the existence and stability of the steady state shock layer solution. In particular, using the projection method, we derived an asymptotic ODE characterizing the slow motion of a shock layer. From this ODE, we designed a specific convergent-divergent nozzle where a, stable steady state shock layer occurs in the convergent part of the nozzle. This is interesting, because it has been proved by Liu [71] and Embid, et al. [34] for the corresponding inviscid problem that shock waves in the convergent part of the nozzle are not stable. We have found that this discrepancy is due to some situations which were not included in [71] and [34] where exponentially small terms, generated by the viscosity term, have to be resolved. Another slowly varying geometry problem is a generalized Ginzburg-Landau equation that can be employed to study the existence of non-constant stable steady solutions to the Ginzburg-Landau equation in several space dimensions. By studying the metastable dynamics in this equation using the projection method, we are able to construct non-convex domains for which the Ginzburg-Landau equation are believed to admit stable spatially-dependent steady state solutions. Chapter 6. Summary and Future Work 160 For the viscous Cahn-Hilliard equation, this thesis gives a detailed analysis of the dynamics of an n-layer metastable pattern and uses a hybrid approach to describe the coarsening process until the final stable configuration is reached. From the D A E system derived from applying the projection method, we can trace the metastable dynamics very accurately. For the Cahn-Hilliard equation (a = 0), using this D A E system, we found the "missing" small terms which are significant for non-collapse interfaces in the O D E system obtained by Bates, et al. [15]. We also showed that the viscous Cahn-Hilliard equation (0 < a < 1) and the constrained Allen-Cahn equation (a = 1) have the same metastable dynamics except in the collapse time scale, while the Cahn-Hilliard equation is quite different. During the layer collapse, a hybrid algorithm based on our asymptotic information and the conservation of mass condition is provided to characterize the whole coarsening process. This thesis has revealed the similarities and differences of the dynamics of an n-layer metastable pattern associated with the three phase separation models in one spatial dimension and showed several interesting phenomena associated with the coarsening process for the first time. The above discussion indicates that our approach based on asymptotic and numerical methods is a powerful and general tool to quantitatively study the metastable behavior associated with various physical problems. Another topic of the thesis is the numerical analysis of a linear boundary layer res-onance problem which is one of the "simplest" metastable models. Our convergence analysis and numerical experiments have shown that several classical finite difference schemes are uniformly convergent with respect to e , but their coefficient matrices inherit the extreme ill-conditioning from the continuous problem. In particular, the exponen-tially small principal eigenvalue does not affect the uniform convergence of these schemes. Thus, the exponentially ill-conditioned singularly perturbed problems would not cause Chapter 6. Summary and Future Work 161 more troubles in numerical computations than other (not metastable) singular perturba-tion problems provided that we could use sufficiently high (such as quadruple) precision arithmetic. Our observations revealed from this model problem were shown numerically to be also valid for some other types of exponentially ill-conditioned boundary value prob-lems and their corresponding time-dependent equations. Thus, these observations might be used as some guidelines in designing numerical schemes for metastable problems. 6.2 Future Research In spite of numerous efforts devoted to study metastable dynamics in various physical problems in the past decade, there still remain many interesting unexplored problems, especially in multi-dimensional domains, and for systems of reaction-diffusion equations. Some of these problems I plan to work on are: 1. Kolmogorov's backward equation in two dimensions, which is related to the exit problem of a Brownian particle confined by a finite potential well. We wish to use the projection method to study the exponential ill-conditioning and metastable dy-namics in this equation and show that the equilibrium solution is extremely sensitive to small perturbations in the coefficients of the equation due to its exponentially small principal eigenvalue. This supersensitivity might be used to give a linearized sensitivity analysis of a model equation that arises from a diffusive regularization of the shape from shading problem. Parts of this work have been done in [103]. 2. Two dimensional flame front problem, modeled by the Mikishev-Rakib-Sivashinsky equation. The goal is to characterize the metastable behavior of a flame front in an axially symmetric vertical channel and to explain experimental evidence showing the slow motion of the flame-front interface. We have noticed an inherent connec-tion between the one-dimensional and multi-dimensional flame front problems, and Chapter 6. Summary and Future Work 162 thus it is hopeful that we will be able to characterize the metastable dynamics by applying the projection method. 3. Two dimensional bubble problem, modeled by the constrained Allen-Cahn equation with a mixed boundary condition. Cahn [23] observed experimentally that a sur-face layer — bubble of the wetting phase continues to exist under certain conditions when this phase is no longer stable as a bulk. This observation does not agree with the simulation of the constrained Allen-Cahn equation with a Neumann boundary condition, since with this boundary condition the bubble is unstable and drifts towards the boundary of the domain. We believe that the resolution of this para-dox is that the boundary condition corresponding to a surface layer perturbs the exponentially small eigenvalue and allows for the stability of the bubble solution. 4. Bubble problem for the Cahn-Hilliard equation. The goal is to characterize the dy-namics of metastable bubble solutions for the fourth order Cahn-Hilliard equation. This would extend previous work which was focused on the dynamics of bubble solutions for the second order constrained Allen-Cahn equation. 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Math. 1 2 0 , (1989), pp. 81-114. [95] G.I. Shishkin, Methods of constructing grid approximations for singularly perturbed boundary value problems: condensing grid methods, Russ. J. Numer. Anal. Math. Modeling, 7, (1992), pp. 537-562. [96] G.I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames— /. Derivation of basic equations, Acta Astronautica 4, (1977), pp. 1177-1206. Bibliography 170 [97] G.I. Sivashinsky, Instabilities, pattern formation, and turbulence in flames, Ann. Rev. Fluid Mech. 15, (1983), pp. 179-199. [98] X . Sun, Numerical Solution of a Non-self adjoint Singular Perturbation Problem on a Nonuniform Mesh. Journal of Nanjing University, Mathematics Biquarterly, vol. 9, (1992), pp. 208-216. [99] X . Sun, Numerical analysis of an exponentially ill-conditioned boundary value prob-lem, in preparation. [100] X . Sun, M . Ward, Metastability for a generalized Burgers equation with applications to propagating flame fronts, European Journal of Applied Mathematics, accepted for publication in Jan., 1997. [101] X . Sun, M . Ward, Metastability and pinning for convection-diffusion-reaction equa-tions in thin domains, Methods and Applications of Analysis, submitted, 1998. [102] X . Sun, M . Ward, Metastable patterns and coarsening for various phase separation models, in preparation. 103] X . Sun, M . Ward, Exponentially ill-conditioned convection-diffusion equations in multi-dimensional domains, in preparation. 104] X . Sun, Q. Wu, A Coupled Difference Scheme for a Nonselfadjoint Singular Per-turbation Problem. Applied Mathematics and Mechanics, 13, (1992), pp. 963-971. 105] T. Tang, M.R. Trummer, Boundary layer resolving pseudospectral methods for sin-gular perturbation problems, S IAM J. Sci. Comput. 17, (1996), pp. 430-438. 106] R. Vulanovic, On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 13, (1983), pp. 187-201. 107] R. Vulanovic, Some improvements of the nonequidistant Engquist-Osher scheme, Appl. Math. Comput. 40 (1990), no. 2, part II, pp. 147-164. 108] M. J . Ward, Eliminating indeterminacy in singularly perturbed boundary value prob-lems with translation invariant potentials, Stud. Appl. Math. 87, (1992), pp. 95-135. 109] M . J . Ward, Metastable patterns, layer collapses, and coarsening for a one-dimensional Ginzburg-Landau equation, Stud. Appl. Math. 91, (1994), pp. 51-93. 110] M. J . Ward, Dynamic metastability and singular perturbations, in: Boundaries, In-terfaces, and Transitions (Michel C. Delfour, ed.) (Banff 95), C R M Proc. Lecture Notes, vol. 13, A M S , Providence, R.L, (1998), pp. 237-263. Bibliography 171 [111] M . J . Ward, An asymptotic analysis of localized solutions for some reaction-diffusion models in multi-dimensional domains, Studies in Appl. Math. 9 7 , pp. 103-126. [112] M. J . Ward, Metastable bubble solutions for the Allen-Cahn equation with mass conservation, S IAM J. Appl. Math. 5 , (1996), pp. 1247-1279 [113] W. Wasow, Linear Turning Point Theory, Springer-Verlag, New York, (1984). [114] M . Williams, Another look at Ackerberg-O'Malley resonance, S IAM J. Appl. Math. 41, (1981), pp. 288-293. A p p e n d i x A Es t imat ing the weight function UJ For e —• 0, we now calculate the weight function u)(x), defined in (2.26), in both the outer and the boundary layer regions. In the outer region we use u£ = x — Xo + t.s.t. to obtain u(x) = exp [-f(x - x0)/e] (1 + t.s.t.) . (A. l ) In the left boundary layer near x — 0 we first integrate (2.13a) to get u'lo(y) exp (- £ f'[-x0 + ulo(z)\ dz^j = u'lo(0), for 0 < y < oo . (A.2) Then, we re-write UJ(X) exactly as u(x) = w(0) exp (-e~l j* f[ue{z)} dz^j . (A.3) Let y = x/e and use ue(x) ~ — x0 + u ; 0 (£ _ 1 x) in (A.3) to get u{ey) ~ w(0) exp (- J* f'[-x0 + uh(z)] dz^j . (AA) Comparing (A.4) with (A.2) we observe that uj(ey)u'lo(y) is asymptotically constant and, hence, oj(ey)u'lo(y) ~ u(0)u'lo(0) (A.5) To calculate UJ(0)U'1q(0) we evaluate the left side of (A.5) as y —> oo using (A. l ) and the decay behavior (2.13b) for u/ 0 . This yields the key identity " ( 0 K ( 0 ) ~ -alovtex?(-f(-x0)/e) , (A.6) 172 Appendix A. Estimating the weight function to 173 where a\0 and v\ are defined in (2.15). A similar analysis, which we shall omit, can be done in the right boundary layer region near x = 0 to show that the product LO(1 — ey)u' (y) is asymptotically constant in this region. The key identity, analogous to (A.6), is that w( 1 -e j /K 0 (y)~w(lK o (0)~a r o i / r exp(-/ ( l -a ;o) /e) , (A.7) where aro and vr are defined in (2.19). A p p e n d i x B Der ivat ion of Equat ion (3.1) Consider the Ginzburg-Landau equation (3.3) with Neumann boundary condition in a cylinder of revolution with cross-section described in dimensional variables by R = R0F (X/L) (see Figure 3.1): Ut = D (URR + R-'UR + Uxx) + QoQ(U) , (B.la) 0 < X < L, 0 < R < R0F (X/L) , ( U R , UX) • (1, -L^RoF' (X/L) = 0 , on R = R0F (X/L) , (B.lb) UX = 0, on X = 0,1 . (B.lc) Here D and QQ are positive constants and Q(U) is described following (3.1). We assume that the cylinder is long and thin so that R0 <C L . In terms of the dimensionless variables r = R^R, x = L~ lX and r = DL~H, (B.l) becomes Ur = S~2(Urr + r~ lUr) + UXX + QQ(U) , 0 < x < 1, 0 < r < F(x), (B.2a) Ur - S2UxF'(x) = 0 , on r = F(x) (B.2b) Ux = 0 , on x = 0,1 (B.2c) Here 6 = i ? 0 / i < 1 and Q = D^QoL 2. We now derive a partial differential equation that is valid for U in the limit 8 —• 0. We expand U away from the endpoints at x = 0,1, as U = UQ + 82Ui + .... Substituting this expansion into (B.2a) and (B.2b) and collecting powers of 82, we obtain Uorr + r^Ur = 0 , in 0 < r < F(x); U0r = 0 , on r = F(x), (B.3) 174 Appendix B. Derivation of Equation (3.1) 175 and Ulrr + r-1Ulr = U0T-QQ(U0)-U0xx, in 0 < r < F ( x ) , (B.4a) UiT = F'(x)U0x , on r = F(x). (B.4b) The first equation gives UQ = Uo(x,t). To determine an evolution equation for UQ we write (B.4a) as (rU\r)r = r(UoT — QQ(UQ) — U0xx). Integrating this equation with respect to r from 0 to F(x) and applying the boundary condition (B.4b) we get Uor = U0xx + 2F- 1F'U0x + QQ(UQ) . (B.5) Let A denote the cross-sectional area of the domain, so that A = TTF 2 . Then from (B.5) and (B.2c) we get the one-dimensional reaction-diffusion equation Uor = \(AU0x)x + QQ(Uo) , 0 < x < 1, t > 0 , (B.6a) UOx{0,T) = U0x(1,T) = 0 . (B.6b) To study the slow motion of internal layers under (B.6) , we suppose Q 1 and so we write Q = e~2 for some e < 1. Then, setting t = e~2r, we find that (B.6) reduces to (3.1) when the cross sectional area A — A(x,e) is given by A(x,e) = 1 + etlg(x)e~e~ld . 

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