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UBC Theses and Dissertations

Regularization methods for differential equations and their numerical solution Lin, Ping


Many mathematical models arising in science and engineering, including circuit and device simulation in VLSI, constrained mechanical systems in robotics and vehicle simulation, certain models in early vision and incompressible fluid flow, lead to computationally challenging problems of differential equations with constraints, and more particularly to high-index, semi-explicit differential-algebraic equations (DAEs). The direct discretization of such models in order to solve them numerically is typically fraught with difficulties. We thus need to reformulate the original problem into a better behaved problem before discretization. Index reduction with stabilization is one class of reformulations in the numerical solution of high index DAEs. Another class of reformulations is called regularization. The idea is to replace a D A E by a better behaved nearby system. This method reduces the size of the problem and avoids the derivatives of the algebraic constraints associated with the D A E . It is more suitable for problems with some sort of singularities in which the constraint Jacobian does not have full rank. Unfortunately, this method often results in very stiff systems, which accounts for its lack of popularity in practice. In this thesis we develop a method which overcomes this difficulty through a combination of stabilization and regularization in an iterative procedure. We call it the sequential regularization method (SRM). Several variants of the S RM which work effectively for various circumstances are also developed. The S RM keeps the benefits of regularization methods and avoids the need for using a stiff solver for the regularized problem. Thus the method is an important improvement over usual regularization methods and can lead to improved numerical methods requiring only solutions to linear systems. The S RM also provides cheaper and more efficient methods than the usual stabilization methods for some choices of parameters and stabilization matrix. We propose the method first for linear index-2 DAEs. Then we extend the idea to nonlinear index-2 and index-3 problems. This is especially useful in applications such as constrained multibody systems which are of index-3. Theoretical analysis and numerical experiments show that the method is useful and efficient for problems with or without singularities. While a significant body of knowledge about the theory and numerical methods for DAEs has been accumulated, almost none of it has been extended to partial differential-algebraic equations (PDAEs). As a first attempt we provide a comparative study between stabilization and regularization (or pseudo-compressibility) methods for DAEs and PDAEs, using the incompressible Navier-Stokes equations as an instance of PDAEs. Compared with stabilization methods, we find that regularization methods can avoid imposing an artificial boundary condition for the pressure. This is a feature for PDAEs not shared with DAEs. Then we generalize the S R M to the nonstationary incompressible Navier-Stokes equations. Convergence is proved. Again nonstiff time discretization can be applied to the S RM iterations. Other interesting properties associated with discretization are discussed and demonstrated. The S RM idea is also applied to the problem of miscible displacement in porous media in reservoir simulation, specifically to the pressure-velocity equation. Advantages over mixed finite element methods are discussed. Error estimates are obtained and numerical experiments are presented. Finally we discuss the numerical solution of several singular perturbation problems which come from many applied areas and regularized problems. The problems we consider are nonlinear turning point problems, a linear elliptic turning point problem and a second-order hyperbolic problem. Some uniformly convergent schemes with respect to the perturbation parameter are constructed and proved. A spurious solution phenomenon for the upwinding scheme is analyzed.

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