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Analysis of a Galerkin-Characteristic algorithm for the potential vorticity-stream function equations Bermejo, Rodolfo 1990

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A N A L Y S I S OF A G A L E R K I N - C H A R A C T E R I S T I C ALGORITHM FOR THE  POTENTIAL V O R T I C I T Y - S T R E A M FUNCTION EQUATIONS By RODOLFO BERMEJO Dipl. Universidad  M.Sc.  Naval  Architecture  P o l i t e c n i c a de Madrid ,1977  The U n i v e r s i t y o f B r i t i s h Columbia,1986  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE  STUDIES  (Department o f Mathematics and I n s t . o f A p p l i e d We accept t h i s  Mathematics)  t h e s i s conforming  to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA January 1990 ©  R o d o l f o Bermejo, 1990  In  presenting  degree  this thesis  in partial fulfilment of  requirements  for  of this thesis for scholarly  department  or  by  his  or  her  I further agree that permission for  purposes  representatives.  may be granted  advanced  It  is  permission.  Department of The University of British Columbia Vancouver, Canada  extensive  by the head of  understood  that  publication of this thesis for financial gain shall not be allowed without  DE-6 (2788)  an  at the University of British Columbia, I agree that the Library shall make it  freely available for reference and study. copying  the  copying  my or  my written  ABSTRACT In  this  thesis  we  Galerkin-Characteristic vorticity  equations  is  stage  a  two  material by  yield  stage.  the  a  grid the  of  the  analyze  integrate  algorithm.  In  the  the  efficient  consists  points  by  of  first  and c o n s e r v a t i v e  The  stage  curves  methods. for  the  this  dependent  interpolation  of  algorithm  the  approximated  algorithm  spline  f o r At = 0(h).  proposed  Particle  updating  cubic  characteristic equations.  and  a  potential  The m e t h o d  the p o t e n t i a l v o r t i c i t y i s  algorithm  the of  component stable  inductive  computationally  an  at  foot  to  Galerkin-Characteristic  Such  variable  method  and  a b a r o c l i n i c ocean.  d e r i v a t i v e of  combining  This  of  develop  the  is  at  advective  unconditionally  The e r r o r  analysis  with  converges  with  2 respect  to  L -norm  order 0(h);  shows  however,  sufficiently  smooth  that  the  algorithm  i n t h e maximum n o r m functions  the  foot  it  is  of  proved  the  that  for  characteristic  4  curves  are  The the  superconvergent  second  Lagrangian  space—time  with  algorithm of  the  is  flow  representation  the  boundary  condition, These other  global  error  and  is  0(h)  estimates estimates  0(h)  for  represent for ii  the  /At). a  onto  projection the  analysis the for  the an  of  Cartesian  coordinated  E l e m e n t s . The e r r o r 2 t o L —norm s h o w s t h a t 2  of  to  of order 0(h  Finite  respect  condition.  the  representation  component  respect  of  Eularian  Crank—Nicholson stage  stage  points  with  for  this  approximation the  free-slip  no—slip  boundary  improvement  vorticity  with  previously  reported  in  the  literature.  The  evolutionary  component  +  K  of  the  2 global that along  error  is  depends the  on  equal the  to  K(At  derivatives  Characteristic.  Since  quasi-conservative  quantitiy,  general  Numerical  small.  theoretical  results  for  h), of  the  one  the  can  the  a  constant quantity  vorticity  conclude  experiments  is  advective  potential  both stages of  iii  where  that  K  illustrate method.  is is  a in our  TABLE OF CONTENTS Page  ABSTRACT  i i  TABLE OF CONTENTS  iv  LIST OF TABLES  vi v i i  LIST OF FIGURES ACKNOWLEDGEMENTS  viii  1  CHAPTER I. INTRODUCTION  CHAPTER II. THE GOVERNING  §1.  Governing  §2.  Notation  EQUATIONS AND THEIR  NUMERICAL DISCRETIZATION  8  Equations  7 14  §3.1.  T h e Weak S o l u t i o n F o r m u l a t i o n  18  §3.2.  The F i n i t e  29  Element Approximation  CHAPTER III. A GALERKIN-CHARACTERISTIC ALGORITHM FOR THE HYPERBOLIC STAGE  38  §1.  Preliminaries  39  §2.  Description of the Algorithm  44  2.1.  The c o n t i n u o u s  2.2.  The D i s c r e t e  Problem  44  Problem  47  §3.  Properties  58  §4.  Error Analysis  60  §5.  Numerical Experiments  70  iv  CHAPTER IV. ERROR ANALYSIS OF THE POTENTIAL VORTICITY AND BAROTROPIC-BAROCLINIC MODE EQUATIONS  81  §1.  Preliminaries  82  §2.  Error Analysis of the Potential Vorticity Equation  86  §3.  Error Analysis of the Barotropic and Baroclinic Modes..102  §4. Numerical Experiments  108  Appendix to Chapter IV  109  CHAPTER V. CONCLUSIONS  126  REFERENCES  130  APPENDIX  135  v  LIST OF TABLES Table  Page  1.  Time E v o l u t i o n o f  t h e Cone  73  2.  Time E v o l u t i o n o f  the S l o t t e d C y l i n d e r  73  3.  Logarithm of  r.m.s.  Errors  112  4.  Logarithm of  r.m.s.  Errors  112  5.  logarithm of  r.m.s. E r r o r s  vi  112  LIST OF FIGURES Figure  Page  1. The Two Layer Model  9  2. Regularization of the Particle Point (X^.t) that w i l l occupied the grid Point x 3a.  at t + At  Rotating Cone after 0 Revolutions  3b. Rotating Cone after 6 Revolutions  74 75 ..76  4a. Slotted Cylinder after 0 Revolutions  77  4b. Slotted Cylinder after 1/8 of a Revolution  78  4c. Slotted Cylinder after 1 Revolution  79  4d. Slotted Cylinder after 6 Revolutions  80  5a. Exact Stream Function Solution at Re = 1000  113  5b. Calculated Stream Function Solution at Re = 1000  114  5c. Pointwise Error distribution of the Stream—Function....115 6a. Exact Vorticity Solution at Re = 1000  116  6b. Calculated Vorticity Solution at Re = 1000  117  6c. Poitnwise error distribution of the Vorticity  118  vii  ACKNOWLEDGEMENTS  I would l i k e  t o acknowledge,  William  f o r allowing  me t h e  freedom  t o roam f a r a n d w i d e i n s e a r c h o f a n u l t i m a t e  scheme;  members  of  my s u p e r v i s o r ,  the following  people;  the  Hsieh,  with gratitude,  my s u p e r v i s o r y  Foreman a n d B r i a n Seymour of  the thesis;  were who  crucial taught  Andrew  On  Staniforth,  me t h e b e a u t y  a more  U r i Ascher,  Mike  f o r t h e i r a d v i c e t o improve t h e t e x t  i n the process  framework o f f i n i t e  committee,  whose  support  o f my r e s e a r c h ;  of doing  and guidance  and John  numerical  Heywood,  analysis  i n the  elements.  personal  note,  whose  companionship  during  the writing of this  made  i t a l l possible,  this  thesis  my t h a n k s  to Martin  and c o m p e t i t i v e n e s s thesis.  Finally,  in particular  i s dedicated.  viii  were  and Alex,  very  helpful  t o my f a m i l y , who  to Maria  Jose,  t o whom  CHAPTER I  INTRODUCTION  During in  the past  t h e r e has been a steady  t h e u s e o f n u m e r i c a l methods  govern  the  ocean  observational scientific study  evidences,  reasoning  from  models  Navier-Stokes equations latter  only  space-time Though  scales  simple,  a  success  is  that  formulation large  of  among  computer a  some  along  with  the  physics  have  very  long  periods  one  the  of  and  a  allow  s m a l l enough  features  time  able in  of  to  to  our  decade.  lead  such to  a  the  reasonably  to minimally  the m i d - l a t i t u d e  to perform  order  of  dynamics.  of  the use of  The  range  the past  which  state  (QGM).  reasons  assumptions,  one i s of  dimensional  significantly  during  important  Thus,  to  to  hierarchy  t o m i d - l a t i t u d e ocean  t h e QG e q u a t i o n s ,  circulation.  This  heat  related  and  models  models  dynamics  the physical  the  mass,  contributed  viewpoint,  of  three  quasi-geostrophic  the ocean  of  technology  hierarchy  the  time steps w i t h a g r i d spacing  resolve ocean  the  QGM  numerical  interaction  use  significant  of  Active  of the ocean dynamics. which  retain  understanding From  to  which  produced  equations  down  the equations  numerics,  has  increase  to solve  dynamics.  d i f f e r e n t aspects  spans  for  25 y e a r s  calculations  study  the  time  e v o l u t i o n o f t h e phenomena. Most o f t h e p r e s e n t models  1  i n the ocean modeling  community  use  standard  The  time  finite  differences,  discretization  leap-frog  scheme  the  possible  by  one  for  is  time  carried  most o f  e x c e p t i o n of  both  the  the  discretization  for  the  viscous  explicitness  the  scheme  without  O(At),  however,  instead of  The s p a t i a l e r r o r of  order  of  main  by  of  and the  where  of  using  this  is  to  significantly  stands  scheme  a  with lagged time  keep  the  altering  the time e r r o r  for  called  are  the l e a p - f r o g  h  so  which  is  of  would  i n QGM w i t h f r e e - s l i p b o u n d a r y  0(h),  space.  the equations,  terms  so  in  terms  purpose  by d o i n g  (At ) as  out  viscous  The  CFL c o n d i t i o n ;  time  terms  step.  of  in  order  suggest.  conditions  generic  the  grid  is  spacing  interval. As of  we w i l l  the  QG  see  in  equations  Chapter consists  equation  -transport-diffusion  vorticity  q  coupled  II,  w i t h an  the  of  mathematical  structure  semi-linear  parabolic  a  equation-  elliptic  for  the  equation  potential  for  the  stream  f u n c t i o n </». It  is  well  known  transport-diffusion the  numerical  equation  is  operator  -  Therefore,  that  equation  analyst.  This  a combination and  the  numerical  poses is  of  dissipative  whether  the  a  due  global  -  o r p a r a b o l i c depends  of  against  term  circulation also  problems  a geographical  to  the  the  other.  the g l o b a l  2  the  -  that the  Laplacian of  the  the  character.  the  such  to an  advective operator  equation  Furthermore, of  of  problem  on t h e r e l a t i v e  structure  d i s t r i b u t i o n or  fact  terms  structure  more h y p e r b o l i c one  challenging  hyperbolic terms  solution  is  magnitude in  ocean  equation  There  .  are,  has for  instance,  narrow  dominant  ones,  significant domain,  away  from  the  efficient  the  the  be  difficulties  understandable area  appear  under  several  [22],  [28],  [31]))  marching  small.  why  in  such  found  with  the  schemes  main  free  of  o f ' these  convective Reynolds  terms  algorithms of  numbers,  dynamics.  The  the  seem  is  in  equations to  be  the  computational  nonsymmetric  l i n e a r approximate operators  for  of  problems which i s  as  t h e new a l g o r i t h m s is  [19],  of  generating dissipative The  main  of  the  flows  at  mechanism produced  unpleasant.  [21],  For  of  the  by  the  instance,  result with a  an e x c e s s i v e l y  high  poorly  small  time  step  the  In  the  so  called  designed  for  dealing  with  such  Galerkin-Characteristic  Method,  b a s e d on c o m b i n i n g t h e method o f c h a r a c t e r i s t i c s w i t h  standard  [31]). as  well  They  the approximate i n t e g r a t i o n . One  a  as  dynamics.  for  dominant  is very  computation  which,  terms  spectrum,  it  an  a  [13],  consequences  very  so,  techniques.  convective  resolved  are  designing  excessive  the  a  spatial  form  objective the  the  And  fluid  i n t r o d u c e d by s t a n d a r d upwinding  emphasis  in  [8],  playing  considerations  algorithms  (cf. [5],  the  parameterized  These  computational  versions  of  the  algorithm.  become  still  areas  boundaries,  very  terms  terms  i n broad  solid  should  convective  diffusive  contrast,  numerical  research  effects  the  transport-diffusion  perfectly  stable  where  with In  terms  out  active  but  role.  dissipative point  areas  finite  element  these papers, convergence  and  procedure  the s t a b i l i t y of conservation  3  (cf. [3],  [13],  t h e method properties  , as have  [27], well been  proved  under  evaluated In  assumption  this  thesis,  the  we  of  the  the h y p e r b o l i c it  has  been  The n o v e l t y purely  of  equation  operator. used  our  for  assuming that  of  the  inner  which  The  in  is  the  grid  at  time  by  t ,  one  dimensional  hyperbolic viewed  first  as  a  Consequently, first  stage  generated At.  On  by  the  progression  space.  stage  particle given  yields the  the In  this  seeks  in  cell  a datum v  :  the  the  algorithm  curves  second via  over  stage the  4  the  the  n  this a  purely can  be  method. hyperbolic  time  involves  stage.  t  into  a mapping  diffusion  s t a r t i n g w i t h the output w from the p r e v i o u s  and  element purely  each  level  algorithm  w = v o X , X being  the  time  procedure  finite m  to  location  constructed  the  -> R ,  characteristic hand,  at  its  respect,  by  of  Explicitly,  particle  Galerkin  generated  an o u t p u t  other of  by  is  the  been covered w i t h a  approximation  projected  researchers.  methods.  Then  is  many  any  combination  -  approximation  new b y  a  and  n  is  means,  c h a r a c t e r i s t i c curve  a particle  we  not  assigns  level  Specifically,  curves  by  Particle  algorithm  to  t h e way we a p p r o x i m a t e  stage  and  algorithm  characteristic  years  20  is  first  the  idea  over  an  the  the corresponding  finite  are  approximate  along At.  products  This form of operating  the c o m p u t a t i o n a l domain has  grid,  the  analyze  equation.  along  algorithm  Galerkin-Characteristic  node  and  methods  hyperbolic  rectangular  propose  inductive algorithm.  transport-diffusion  for  all  transport-diffusion  a two s t a g e  typical  of  that  exactly.  integrate use  the  of Q step  a  time  mechanism, The  output  v of  this  next  time  time  projection  into  second level.  the  parabolic  of  with  Euler  the  any  the  the input f o r  second  standard While we,  of  difficult  nor  conditions  algorithm  of  the  expensive.  to  be  the  for  for  discussed  in  of  are  two  the  Chapter  as  the  the  flow  scheme  for  employ  the  reasons.  to get,  in  First,  in  stage,  scheme  IV,  the  interested  hyperbolic  we w i s h  at  representation,  authors  Crank-Nicholson Second,  of  discretization  contrast,  scheme  our  thought  Eulerian  of  in  be  stage  representation  time  most  the f i r s t  may  space-time  Crank-Nicholson  context  stage  Lagrangian  scheme,  implementation  in  The  equations.  backward  the  is  Cartesian  coordinated  using  stage  the  is  neither  under  certain  a higher  accuracy  time. The  solved  spatial is  d i s c r e t i z a t i o n of  made  by  functions  on a g r i d  of  this  using  purely  C°-finite composed  kind  hyperbolic  of  constructed.  We show  the upstream  l o c a t i o n of  spline elements,  polynomials clear  how  integrals  as  for on  the  With  example,  can  of  the  get  elements as  the  a  linear  bilinear  The m a i n is  is  on  1  be  basis  at  the  solution  is  solution  at  to  cubic  of  C°-finite  triangles,  quadratic  elements,  relationship  products  and  evaluation  of  algorithms.  the  G a l e r k i n - C h a r a c t e r i s t i c method  5  apparent  classes  to  advantage  equivalent  interpolation classical  The  with  particle  C -finite  simple  inner  equations  projected p a r t i c l e  other  or  of  rectangles.  the p a r t i c l e s  rectangles,  one  of  stage  that  interpolation.  elements  finite  first  the set  it  is  not  between  the  simple such  linear  integrals  proposed  in  in [13]  and  [31]  is  eventually have  a  very  lead  demonstrated.  approximation  t h o u g h no  show  in  [27]  ideas  exists,  Chapter  compute  III.  related the  still  In  curves  meteorological  literature  use  of The  briefly  thesis  first  III  performance Chapter  2 L -norm  and  of IV  of  integration  as  follows.  In  the  on  of  the  in  for  numerical  a  the  formulation.  analysis  of  order  illustrate  to  estimates the  with  the  numerical the  equations.  respect  algorithm  vorticity  we  two-layer  in  potential  the  II  Some  of  the  heuristic  algorithm.  stage  that  of  the  the e r r o r  second  to  the f o o t  the a l g o r i t h m w i t h l i n e a r h y p e r b o l i c we g i v e  used  Chapter  the f o r m u l a t i o n and  presented  in  [34].  their  of  that  theoretically  construct to  we  mention  based  substantiate  as  classical  utilized  QG e q u a t i o n s  stage  are  previously but  integrals  are  also at  our  interpolation  the  should  [27]  apply  mention  of  continuum  devoted  hyperbolic  experiments  In  the  ocean is  organized  can  may  al  algorithm,  we  i n t e r p o l a t i o n made i n  is  introduce  stable  [34]),  we  et  linear  the v a r i a b l e s  (cf.  Morton  methodology  We  been  thesis  cubic spline  baroclinic Chapter  this  has  which  we  connection,  method.  i n t e r p o l a t i o n of  In  and  a  [19]),  inner product  products  characteristic  arguments.  get  particle  inner  Galerkin-Characteristic cubic spline  the  them  this  with  as  Nevertheless,  between  and  (cf.  algorithm  to evaluate  relationship  algorithms  process  t o an u n s t a b l e  recently  particle  costly  to in  equations.  the the  Also,  2 L -norm  error  estimates  and v e l o c i t i e s a r e  of  the  studied.  6  approximate  stream  functions  Each  chapter  the s e c t i o n s of  divided  into  sections  and,  when  needed,  The  subject  are f u r t h e r divided into subsections.  each chapter  technical  is  is  i n t r o d u c e d by a s h o r t  details,  of  the  main  description,  aspects  and  free  results  of  of the  chapter. The  criterion  chapter  consists  f o l l o w e d by  denotes  refer  to  of  have  are also  proof  appendix,  writing  equation  first  attached  the  an  number  first  of  been  a  Chapter of  Lemma  placed at  IV  2.3  is  of  the  If  chapter  V.  Two  we  deals of  t h e end o f  with  this the  theorem.  7  The  the  do  it  Appendices  the  we h a v e  throughout  the  by  Conclusions  first  In  to  number  one  technicalities  chapter. thesis,  Thus,  we h a v e  technical  completeness.  each  section  such section.  closed w i t h the  i n e q u a l i t i e s which a r e employed  the Lax-Milgram  number  in  the c h a p t e r f o l l o w e d by the  for and  equations  s e c t i o n 3.  different  Chapter  included  the  12 o f  The t h e s i s in  the  the e q u a t i o n i n  t h e number o f  presented  to  to  e q u a t i o n number  the equation.  which  the  of  t h e number o f  (3.12)  writing  adopted  is of  second  collected thesis  and  CHAPTER II  THE GOVERNING EQUATIONS AND T H E I R NUMERICAL D I S C R E T I Z A T I O N  This  descriptive  equations them.  and  In  the  proposed  addition,  throughout  the  chapter  we  thesis,  introduces  numerical  also  present  and  the  the  algorithm the  spaces  basic  where  continuum  to  integrate  notation  the  used  solutions  are  sought.  1.  Governing  We w i s h the  Equations  to apply the proposed n u m e r i c a l a l g o r i t h m  c i r c u l a t i o n of  to f a c i l i t a t e example two  of  layers  a mid-latitude baroclinic  the mathematical a n a l y s i s  such of  an  ocean.  Thus,  densities  mid-latitude,  our  and p  6-plane,  ocean.  we c h o o s e  ocean  rectangular,  study  In  order  the  model  respectively  2 >  to  simplest  consists  confined  flat  bottom  with  solid  of  to  domain  a D  A rotating Figure  with 1  angular  illustrates  coordinate  system,  the  The  model.  each  layer  subindices upper  and  velocity  stream  are 1 and lower  as  the well  as  the  and  for  whereas  8  the  of  geometrical  and  by  stand  layers,  arrangement  functions  denoted 2  Cj a n d  relative u ,  Cartesian  parameters  of  vorticities  of  respectively.  magnitudes  1- s t a n d s  our  boundaries.  for  related the  to  The the  magnitudes  Fig.  1 The Two L a y e r Model  9  evaluated at The acting  flow  on  through  is  driven  free  in  the v e r s i o n  by  variable  The  of  wind  coupling  the  stress  between  interface.  We  [20].  The  z-dependence  central  Thus,  defining  finite the  of  T(t,x)  layers  use  potential vorticity-stream  employing  direction.  each l a y e r  a  surface.  c a n be f o u n d i n  vertical  by  displacement  modelled  we  interface.  is  the  the  equations which  the  the  the  QG  function variables  differences  potential  is  in  the  vorticity  at  by  have  i  2  Dq  -fif-  - A V'q. H  where Q D Dt f  =  T  df/dy  - 5.^  2  qAx,t)  =  qXx.O)  = f  = Qx[0,  = 0,  on t h e  i n <?  r  , i  boundary  =1,2,  (1.2.1)  T,  -  ,  (1.2.2) (1.2.3)  T],  8 dt 2\Q\sin9Q  approximated its  + S. cq.  tangent and  =  f  + (3y  is  the  Coriolis  by p r o j e c t i n g the e a r t h ' s plane  at  the  reference  are parameters  parameter,  spherical  latitude 6 =  whose m a g n i t u d e s  10  which  geometry 6^,  and  depend on  is onto /3 =  | f i | , Qg  and t h e r a d i u s g'=  of  (  the e a r t h  )g, p  here  g  (cf.  [30]).  denotes  the  acceleration  due  to  2  gravity, _ . the f o r c i n g F  curlT  =  H  A„ i s ti  is  a  over  two  Define  layer  *(x,t)  ocean,  = $  H  V(x,  barotropic we  may  =  t)  and  modes,  as  - ifi (x,t)  and  \jt  lead  2*2 „  +  ,  2  (1.3.1)  H  $(x,t)  ,  are  (1.3.2)  the  respectively. decompose  so  called  According  the  baroclinic  to McWilliams  baroclinic  mode  *  in  and [26], the  fashion  a  considerations  the g l o b a l  of  follows.  ¥ ( x , t ) = * (x, t) + C(t)V By  combinations  e q u a t i o n s w h i c h a r e now f o r m u l a t e d .  (x,t)  l*l '  further  following  linear  and $ ( x , t )  t)  *(x,t)  assumed  symbol.  of e l l i p t i c  *(x,  c o e f f i c i e n t which i s  l i n e a r bottom f r i c t i o n c o e f f i c i e n t ,  the Kronecker  to a set  where  eddy v i s c o s i t y  t h e w h o l e d o m a i n Q,  a constant  8. . i s For  '  the l a t e r a l  constant c  l  of  s  (x).  mass c o n s e r v a t i o n  (1.4) [26],  ^(x,t)  satisfies  condition  J  f fi  Kx,  t)dQ  = 0 , V t € [0,T]  11  .  (1.5)  In view o f  (1.4) and (1.5) the time dependent  determined a t any i n s t a n t  (x,  of  (1.1),  CV  2  , V t e [0,7].  2  (1.6)  (x)dn  ( 1 . 3 ) a n d ( 1 . 4 ) we o b t a i n  - A ; * (x) s  = 0 i n Q_ , 1  2  *  CV  t)da  = - — f *  By v i r t u e  is  t by  * C(t)  function C(t)  s  (1.7.1)  |_ = 1 , V t . r  - A ;* Cx,t)  = q  2  a  *  a  |_ r  2  (1.7.2)  - q  in Q  2  T  ,  = 0 , Vt ,  (1.8.1)  (1.8.2)  where  f r// +ff; 2  V *  =  *  3  ffq  = V  #|  Vt .  The s y s t e m s ( 1 . 7 ) - ( 1 . 9 )  *  or  a  (x,0)  ( i  - 8  3 )  + ff g  vVx,0 = 0  •  2  A  ii  - / i n Q_ ,  (1.9.1) (1.9.2)  are closed with the i n i t i a l  = $(x,0; = 0 in Q ,  equivalently  12  conditions  (1.10.1)  ip^x.O)  Finally,  = \p (x,0)  the  velocity  d e t e r m i n e d a t any  u  1.2. The  = curty  i  stream along  On  ,  and  the boundary  other  T;  at  that  hand,  ,  the  is  for  As  vorticity  formulation  well  vorticity  physical  however,  the  of  n  the boundary,  t  instant  meaning  transport in  the  boundaries,  r  state  that  t,  the  constant  ( 1 . 12)  the  N-S  is  an  solid  denoting  the  the is  that  across  a  the  solid  value  of  the  boundaries  of  the  of  the  boundary  assuming  condition  that  ,  normal  the  function-relative  ( 1 . 13)  and  tangent  r e s p e c t i v e l y the f o l l o w i n g r e l a t i o n s  13  boundary  from the values  additional  Specifically,  = g.(x,t)  zero  equations, the  of  mode  stream  one can e a s i l y deduce,  the  and  any  barotropic  unknown"on  the stream f u n c t i o n .  with  (1.8.2)  i = 1,2.  known  is  u.l  and  is  is  = C (t)  r  boundary.  for  layer  (1.11)  ip^ a r e ,  i n t e g r a t e d normal  on  each  .  and  vertically  velocity  at  relation  (1.7.2)  (1.9.2)  domain;  (u^,v.)  i = 1,2  condition  relative  u^=  Conditions.  functions  the  (1.10.2)  t by the  conditions  <p.\  in  vector  instant  On B o u n d a r y boundary  = 0  2  vectors hold  on  30.  — |_ = -g.ot  Sn  T  ,  "i  (1.14.1)  dip  at  If in the  * other  terms, In  on  the  free-slip  if  wall,  so  boundary  *  stress  on the s o l i d w a l l ,  relative vorticity is  and 0 is  =  0,  known  conditions  for  then  there  exists  on  the  Now,  the  case  g^  boundary  is  on  generated.  = 0  is  =  as  denoted  the value i  A ,  or  shear  literature  assuming that  the p o t e n t i a l v o r t i c i t y  0 - -j— C(t) g n .  is  i n f l u i d dynamics  condition.  vorticity  i d e n t i c a l l y zero  relative vorticity  condition while  boundary  relative  i s no s h e a r  contrast,  boundary  by n o - s l i p the  the  condition  (1. 14.2)  6  then there  wall.  stress The  0,  — |_ = g.on = 0 . T i  1,2  of the  are  f  q.. ul  2.  = X  + f  1  , V t e 10,TL  (1.15)  Notation  We  now  throughout  introduce the  some  thesis.  Let  basic fi  be  notation an  open  which  subset  is  of  R  used  2  with  2 Lipschitz space.  Let  which have  boundary C^Cfi;  T, be  continuous  w h i c h a r e bounded  R  is  the  set  the of  two real  dimensional valued  partial derivatives  o n fi. On C^(Q)  14  of  we i n t r o d u c e  Euclidean  functions  order  at  the  norm  on  least  fi k,  \v\.  = '  k  Boldface  symbols  sequences;  sup xeQ \a\<k  ' °  M  \D v\  denote  either  the meaning w i l l  multi-index  notation.  (2.1)  .  a  If  matrices  ,  vectors  be c l e a r f r o m t h e c o n t e x t , IN  denotes  the  set  of  a  or is  a  nonnegative  integers  a  We h a v e  :=  ( a , , a „ , . . . . a ) , a . e IN, 1 2 n l  1 = 1,2  . . ,n  the f o l l o w i n g d e f i n i t i o n s  |a| = a . + a + . . . + a , 1 2 n a £ (3 i f f (a.+ ( a  =  D  a  a  > Vi  i  = a  i~  i  0  +  i  ~  1 8  i  (a'.)(a>) 1 2  Ca - &) x  i  |3J  a/ =  - P  a  d  o n lP(Q),  i  = 1,2, . . ,n =  3x  ,  iiuii  !)  ,  ; 0)  , Vi  = 1,2,  . ..,n ,  an n  1 s p < co ,  the space  measurable  of  all  equivalence  functions.  The  norm  d e f i n e d by  = p  '  2  r e a l - v a l u e d Lebesgue p < co i s  ,  ' '  3  n  a  ai  We d e n o t e b y LP(Q) of  V  ,  , an. (x ) , n  3x, 2  classes  '  - p.  , a i , , a.2.. = (x, ;(x_ ; 1 2  «  , Vi  ...(a  = max(a.  i  = 1,2, . . ,n  n  ( { l"l <*n J p  15  P  ,  (2.2)  with  Hull  When  p  =  = ess  2 we d r o p  sup\u\,  Vx € Q .  the subindices  p  and Q  if  there  is  no  confusion. For  e a c h i n t e g e r m 2: 0 a n d r e a l  Sobolev  spaces  W ' (Q)  =  m P  (Q),  p,  1 < p < 00 , we d e f i n e t h e  as  v e L I T Q ; : D v € L (Q),  j  P  l a l s m . l s p s . o J .  P  a  (2.3) T h e s p a c e M '(Q)  i s a Banach s p a c e w i t h t h e norm  m,P  Ml  n m  '  P  '  = (  Q  £ IID°Vll \a\sm  _ ) P  '  , 1 s p < 00 ,  1 / p  (2.4.1)  Q  and t h e seminorm  |  V  » O = <  L  I  l^ln  '  O  (2.4.2)  When p =00 llvll  Also,  _ =  the space  represents  max (ess I a I sin  W^^Cfl.)  sup\D%\),  is  the completion of  separable t h e space  Vx e Q .  for  1  (fen)  (2.4.3)  < in  p  <  00 a n d  t h e norm  II II m,p,Q Let  X  be  a  Banach  space  with  norm  r e p r e s e n t s a f u n c t i o n d e f i n e d o n Q^, we s e t  16  II  II.  If  v(x,t)  L (0,T;X)  =  P  | v:  | \\vl\ dt  < oo |  P  (2.5.1)  ,  1/p = (T \\v\\ dt VO  Ivll  P  L (0,T;X) P  (2.5.2)  T 1^(0,  T;X)  = |  v:  J  di  IID*vll'  m  < oo, a ^ m  _ -\ 2/2  llvll  {/"(O.TjX) a h e r e D.denotes '  a 3 t  (2.5.4)  a=0  a  a  ess supWD v\\ < oo , a ^ m j - , f 0, 17  Hull  = max ess supWD v\\ , m Ottm [0,1]  ^'"[O.TjX] In  (2.5.3)  0  (2.6.1)  (2.6.2)  particular  L (0,T;X) P  = H°(0,T;X),  L (0,T;X)  = W°' °°(0, T;X).  m  S o m e t i m e s we u s e t h e s h o r t h a n d n o t a t i o n ,  i f c o n f u s i o n does not  arise,  L (0,T;X) P  For  p = 2,  s L (X), P  IT^'^n;  iT(0,T;X)  =  ^(X).  = [/"(n), w h e r e lF(Q)  o f o r d e r m. The s u b s p a c e H^(Q)  17  of lf(a)  is  the H i l b e r t  i s d e f i n e d by  space  = 0, 0 £  3. T h e Weak S o l u t i o n  We  formulate  problems with,  (1.2),  in  this  section  (1.7)-(1.10)  we d e f i n e t h e c l a s s e s  H^(£2);  subsets  however, of  solutions  it  H*(n)  (2.7)  m-2  Formulation  f o r m u l a t i o n o f t h e weak is  I<x| ^  t h e weak  and t h e v e l o c i t i e s u^.  convenient  which  are  we a r e s e e k i n g .  of To  of f u n c t i o n s which a r e used  form of the problems. is  solution  specifically  begin i n the  Our b a s i c  to introduce  some  related  the  space closed  to  the  Such subsets a r e :  (3. 1.1)  (3. 1.2)  Thus,  the stream f u n c t i o n s  i/». € S , w h e r e a s t h e p o t e n t i a l a n d l c . v o r t i c i t i e s b e l o n g t o S^. T h e s t e a d y c o m p o n e n t * (x)  relative of  the b a r o c l i n i c  other  hand,  the  mode  time  V(x,t)  is  a n d t h e b a r o t r o p i c mode $(x,t) In (1.2),  order we  developed the  to  motivate  introduce i n Chap.  l i n e a r -advection  w i t h C = 1. On t h e c b a r o c l i n i c c o m p o n e n t * (x, t) 3.  dependent  III.  a r e i n HQ(Q) f o r a n y t €  the  here  in S  algorithm  some  Recall  equation  18  ideas  to  solve  which  equation  are  the c h a r a c t e r i s t i c  [0,T].  further  curves  of  at  (3.2.1)  q(x,0)  satisfy  =  q (x), Q  t h e system  dX(x,S;T)  = U(X(X,S;T),X)  dx  Xix,  ,  (3.2.2)  =x ,  s;s)  w h e r e u i s t h e v e l o c i t y v e c t o r w i t h d i v u = 0. The  transformation  conditions  to  be  transformation equal  (cf.  of  t o 1 almost  To  solve  -»  into  defines,  X(x,s;t)  specified  fi  below,  itself  with  under  a  certain  quasi-isometric  Jacobian  determinant  everywhere.  the potential  vorticity  equation  ( 1 . 2 ) we s e t  [31])  ^  = ^ (x,t:T),r)\ (X  where  denotes  The  hand s i d e  q  x  right  along - the  consideration  the total  derivative.  of (3.3) represents  characteristic  suggests  (3.3)  ,  T=t  the  curves  following  algorithm. Suppose that a p a r t i t i o n o f  19  [0,T]  the time e v o l u t i o n of of  the  two-stage  flow.  This  inductive  ? •• = \ 0 = t  is  <t  Q  s p e c i f i e d . We s h a l l  by T upon  is  define  Q -» R , i  m  =T1  the semi-discretization  induced  0 s n £ m, we a s s u m e  = 1,2,  2  that  a ' r e f l e c t i o n ' map  E : H (Q) n L (Q) S  such  < t  i n t h e f o l l o w i n g two s t e p s :  (1.2)  1) G i v e n q\: there  <  t  that  > H (Q' ) r\ L (n' ) ,  a  S  functions  C^(Q)  a  a r e mapped  onto  functions,  C^(Q')  and  £  i s l i n e a r and continuous,  a n d Q' i s d e f i n e d b y  n+1 n  1  '  * Thus,  2  a n y f u n c t i o n q : Q' q.  Consequently,  > R c a n be e x p r e s s e d a s  =Eq.  3.4  t h e o u t p u t from s t e p one  i s then given by  5? = S ^ V * ' W V ' V . The  method  this  thesis,  chapter.  to approximate  f u n c t i o n q^.(X,.(x,t  ^hi  q } i s one o f t h e c o n t r i b u t i o n s 1  and i t w i l l  F o r now, i t  hi  approximate  values,  space  where  be d e s c r i b e d  is  sufficient  , ; t ), t  n+1  does  (3.5)  n  n  ),  the solutions  20  to  where  not belong  in detail  i n the next  indicate  that  the subindex  to a finite  a r e sought;  of  the  h denotes  dimensional however,  the  f u n c t i o n q\ . m u s t  belong  make t h e f u n c t i o n a },  be i n  1  hi  from  other  notation  t o H,.  It  i s the fashion  t h a t makes o u r m e t h o d d i f f e r e n t  h  Galerkin-Characteristic  i s now i n o r d e r :  methods.  is  1  X.(x, l  t  2)  computed  t  n+1  from  1  such  n+1  l  q (x)  1  i2~^~  A  q  (  i  + i  1  t  To  (3.5),  that q -  2  = A  n  solve  g"  n+1  ^i  +  J  bi  ,  e  9 / )  + f  =  -  ) +  5  i  - 4 - f v<rgf  /  f  n  - r ^ - c "  i  the boundary  +  '  Z  /  2  '  Q)  in T  values  i  points  2  such  n  e T  >  ,  = HQ(Q) ® M .  that Vu e W  21  ve ;  + Ztf)),  1  V  i  G  =  H  oC ^ n  (3.6.1)  (3.6.2)  1,2  ,; t  n+1  e  ).  n  A." o f t h e r e l a t i v e v o r t i c i t i e s  t h e method o f G l o w i n s k i  ff rn;  and Vr  0  and c o - w o r k e r s  L e t M be a complementary subspace  +  the  the following:  e HICQ)  7  Find ^ " ^ e  at  that  u  Q  t/l we a d o p t [16]).  takes  1  = X_uvdQ , and X ? = X.(x,t  (u,v)  update  g?  on  A  1  " V * ^ '  +  g"t  where  i  ^1  values  - g.CX")  n+1  d  the  Given the output from  i  word  ).  n  Find q ,  ( -  A  We s e t g ? = g.CX ?,) t o r e m i n d u s ^1  g?  i n w h i c h we  o f H^(n);  (cf. [11], i.e.,  (3.7.n  lfj - a " 1* ,  n) = ( V 01" ,  1  Next,  V  +1  n+1  ; - J  M  u ds .  (3.7.2)  we calculate the barotropic and baroclinic modes, the  stream functions and the velocities.  i)  Find 9 (x) e S ,c = 1, such that V<£ e HI(Q) s c 0  CV* , 70 ; + A ( * . <f>) = 0.  (3.8.1)  2  s  ii)  Find <b (x),  $  n+1  a  n + 1  (x)  e n\(Q) such that V0 e fl^ffi;  C/  c  /n.n+l „. . . 2, n+,l ,. , n+1 n+1 , (7* , 70; + X (9 , <(>) = -(q - q_ ; , 3 3. 1 c.  ,_ „ (3.8.2)  T  (V9 ,  V<f>) = -(b ,4>)  n+1  ,n+l b  where  ,  n+1  H  =  „ n+1 ., n+1 l l 22 q  +  H  H +H 2  (3.8.3)  , - f  q  2  iii) f ^  From  (1.4),  * (x)dn n+1  = - - ^ 1 * (x)dQ  (3.8.1),  (3.8.2)  .  and  (3.9)  (3.9)  we  are  able  to  determine the baroclinic mode ¥ * ( x ) by n+  V  n + 1  (x)  = <H (x) a n+1  + c"" " * (x) . s 1  (3.10)  1  Once the baroclinic and barotropic modes have been calculated, at  instant  t ,, n+1  the  stream  22  functions  1  and  2  are  obtained  i n view o f (1.3.1)  .n+1  *  2  ,n+l  " 0  by s o l v i n g  n+1  = *  2  n+1  and (1.3.2)  n+1 n+1  H+H  Finally,  u  (3.11)  2  n+1Cx)  i s obtained  from  n+1 , n+1  n+1,  i  Equation  Remark.  beginning Our  (3.12)  (3.8.1)  once  next  admit  shall  assume  objective  ourselves  of  t h e systems  i n H*(Q),  f o r any t  that  t h e boundary  values  are  sufficiently  smooth.  prove  the existence  ( 3 . 6 ) f o r t h e upper  proof  f o r the lower  main  tool  used  in  layer  spaces  V ' (Q) m P  (T),  Towards  is  T.  We  restrict  and uniqueness  of the  layer  Also,  (i =  1), since the t h e same. T h e  the Lax-Milgram  t h i s end l e t us f i r s t P  —m W  theorem introduce  and t h e trace  W^' (Q),  where T i s t h e boundary  space  e  and  we  o f t h e spaces  1 £ p < co, t h e d u a l  (3.6)  and t h e f o r c i n g  ( i = 2) i s e s s e n t i a l l y  the proofs  ( c f . [2] and A p p e n d i x ) .  spaces  that  solution  to  solutions  i s t o show  a uniqe  function  For  and f o r a l l a t t h e  o f t h e computations.  (3.8)  the d u a l  i s solved  o f fi.  D'  is  (Q)  t h e space  of  0 continuous  l i n e a r f u n c t i o n a l s d e f i n e d o n w"l' (Q) w i t h t h e n o r m P  23  IfII  where p ' It  is  ,  =  0  satisfies  usual  to  p  if  1  J  1 / 2  -1  -1  + p'  [  <  ' " „ „ Hull _ in, p, Q f  >  = i , and  space  (3.13.1:  , u * 0 ,  l  r  <f, u> =  <f,u> a s d u a l i t y  denote  belongs to the dual The  sup ,m,p uelt^j  -m, p ,ii  fudfi .  pairing,  where  f  o f u.  t r a c e spaces a r e d e f i n e d by  = j <7 e L (Q)  (D  ; 3 u € ^(Q)  2  fa  yyx = g  on V J ,  f o  gj where y . =  dn  J  — i s the trace operator  of j - t h order  which can  J  be e x t e n d e d  to continuous  (D.  linear  operators  mapping  I n t h e f o l l o w i n g we u s e t h e s p a c e  H ( Q) m  (D  H  onto with  norm d e f i n e d b y  '•7J  ,,o  1  o r  b m-1/2, 2, T  =  i  n  f  im n •  l l u l 1  (3.13.2)  uelf (Q) >2>& 1  Remark. F o r f u r t h e r p r o p e r t i e s o n t h e t r a c e o p e r a t o r s a n d t h e s p a c e s H" ^ ^^ (T) 1  Proposition  see Appendix  2  For q ^. €  3.1.  1  1  of Chapter  H  1  /  2  1/2  Proof.  Since  exists  a q^ e H (Q) s u c h  Next,  H  (D  X  consider  is  the range that  the b i l i n e a r  form  24  and  ( D  problem (3.6) has a unique solution  IV.  F  X  I  +  1  /  Z  €  H~ltl),  in H*(n). of  1  there  o n T,  f o r i = 1, 2.  i n H (Q), t h e n  1.  = c j ^  u, v e HQ(&) >  Ku.v) = (u,v) + v(Vu.Vv), LtA where It  is  well  known  coercive; for  H  v =  that  (cf.  is,  [29])  that  there exist  \I(u,v)\  constants  < C  Hull  \I(u,u)\  Also,  is  I(u,v) the  problem  subindex  i)  n+1  that  £ C  n+1  continuous we h a v e  to  functional;  n+1/2  to  examine  for  i  if =  1  - q (x)  nfl  Q  to  ,  e) + (F  ,  n+1/2  u,v e H*(Q). (we  drop  the  e H^(Q) a n d  e) . ve e H^Q)  apply, the Lax-Milgram  theorem  (3.14)  defines  i.e.,  > (q(x"),  (F  a n d H*- c o e r c i v e  Q  the r i g h t hand  belongs  .  2  1  0  8  0  H  e H (SI) s u c h t h a t q (x)  - Kq , order  Hull  0  q , B) = (qCx"), B) - vCVqCx"), V8) -  I(q -  so  > 0 such  HQ-  is:  F i n d q (x)  that  ,  and  Hull H  In  continuous  v i n Hg(Q)  u,  Now,  is  I(u,v)  side  the  of  it  . (3.14)  remains  to  a continuous  show  linear  mapping  8) - v((Vq(X ), n  VO) - I(q , 8) + ( Q  \ )  T1+1/2  F  Q  H *(&). We f i r s t n o t e t h a t Kqg> 0) i s c o n t i n u o u s , 8) - I(q , Q  B) e H~ (Q). 1  inequality  25  Next, . by  the  Schwarz  vCVqOf"), VQ) £ C\\Vq(X )\\ IIV6II , n  t where C i s the  a constant  and X  quasi-isometric  Theorem 1.1.7  of  is  in  , which  homeomorphism  [25]  x  —>  is  the By  X. n  image  of Q  virtue  of  and t h e f a c t  \\q(X )\\ = \\q \\ , n  since  the  Jacobian  homeomorphism HVqll i s  is  the r i g h t With  satisfied,  the  follows  •  P r o p o s i t i o n 3.2.  in  in HQ(Q),  we  have  that  (3.14)  is  and  ,  i n H ^(0.).  of  the  uniqueness  Lax-Milgram of  q ^(x) n+  theorem  H^(Q)  e  (3.8.2) and (3.8.3) have a unique  whereas the unique solution  of (3.8.1)  lies  ^(Q).  Proof. before, the  2  assumptions  Problems  quasi-isometric  everywhere,  v e ; e H~ (Q)  1  existence  the  Hence  e) - vdqof ),  the  of  1 almost  n  hand s i d e of  all  solution  to  t o IIVq ll.  (qCx"),  so  determinant  equal  equivalent  n  We f i r s t p r e s e n t let  Problem  e H*(Q) (3.8.1)  Find *  s  the proof such  for  that  i n the f o l l o w i n g  - *  0  0  € HI(Q)  0  26  the Problem 1 o n T.  (3.8.1).  Next,  fashion:  s u c h t h a t V</> e  H\(Q)  0  we  As  recast  w* - v ), v^; + Q  A C* -  * ,  2  S  0) = -cv*, 0  Vc5; - A  0)  ^ ,  2  (3.15) Define  now t h e b i l i n e a r f o r m I'(u,v)  = CVu, Vv) + \ (u,  I'(u,v)  I' (u,  (3. 15) b e l o n g s guarantees Next,  a n d H*- c o e r c i v e ,  t o H *(£l).  Therefore,  we p r o c e e d of  essentially  t o prove  problems  t h e same  i n H*(Q).  the existence (3.8.2)  hand  side of  theorem  of the solution  V (x).  and uniqueness  and (3.8.3).  f o r both problems.  Secondly,  so the right  t h eLax-Milgram  t h e e x i s t e n c e and uniqueness  solutions  are  v).  2  i s continuous  v)  as  of  The p r o o f  First,  note  t h e b i l i n e a r f o r m I"(u,v)  the is  that  cr? i +  defined by  = CVu, Vv) + u C u , v) , u, v € H (a),  I"(u,v)  Q  2 where  p. =  (3.8.3),  A  is  assumptions  f o r Problem  continuous  (3.8.2)  and u  a n d HQ(Q)-  o f the Lax-Milgram  =  0  coercive.  f o r Problem  Since  theorem a r e s a t i s f i e d ,  a l l  the  then i t  1. follows *  t h e e x i s t e n c e and uniqueness  i n HQ(Q).  3.2 and (3.11)  and u n i q u e n e s s o f t h e stream s u b s e t s S„ a n d S „ . o f H (Q),  l  H  1+  H  2  '  C  2  the existence n  1  = ~0-nr  2  27  one deduces a  °2  are defined by  =  and  f u n c t i o n s "A"*^ d ip^^in t h e r e s p e c t i v e l y , w h e r e C , a n d C_  1  °1  ¥  •  From P r o p o s i t i o n  C  of the solutions  a  n  d  ^  ^  given  2  by (3.9):  Remark On Regularity With  reference  Lax-Milgram  II Hq  of the  to  the  Solutions.  regularity  theorem s t a t e s  of  1,2,0  11 i  < c ii »  ll* |l a i,2,n  ~  4  n  w h e r e Cy  that  g  n  1,2,0.  n 2-1,2,0 '  s C lib" -1,2,0 6  C^,  and  are constants  w h i c h d e p e n d o n Q.  2 However,  one c a n assume  s c a l e wind  condition  qAx.O)  regularity relative  by  i s a smooth  the regularity  In f a c t ,  vanishes  at  suppose theory  that  .  for elliptic  tha  and t h e i n i t i a l polygon  to expect  i f one assumes  t h e boundary, g  because  Q i s a bounded  i t i s reasonable  of the solutions.  we m i g h t  function,  Since  m  corners,  2  F e L (0,T;L (0)),  = f e C (0).  vorticity t  that  stress  no r e e n t r a n t  instant  the  :  " II* 1i s 1,2,0  ll* |I  with  solutions  n  n  large  the  then  = f + c" € H problems  higher  that the a t any (T) a n d  (cf. [17])  we  have " ^ " 2 , 2 ^ ^ 7 - !  where  the constant  " " F  Ai"3/2,2,0  +  depends  \>  (3  - 16  o n 0.  When t h e r e l a t i v e v o r t i c i t y d o e s n o t v a n i s h  a t the boundary  is  3/2 not  c l e a r w h e t h e r q^ e H  f r o m o u r f o r m u l a t i o n t h e most 2*2 2 /2 i s t h a t g^e H (T); t h e r e f o r e ,  (D;  we c a n s a y , i n v i e w o f ( 3 . 7 ) ,  28  n  by v i r t u e  o f t h e t r a c e theorems  q Next,  we  examine  sufficiently that  from  ll+2  e H ^),  the  then  w i t h no r e e n t r a n t  there  our problem corners,  number  is  a  If k  equations  fi  £ 0  "•"'W^n  x  3  two d i m e n s i o n a l  polygon  ( 3 . 16. 3 )  k  9  c  i s g i v e n by  ( k  >  * r*2 k.2,n'  a n  n  n  X.2,n  a ) n b  •  t h e n ¥ , $ . e H^(&); n  1  imposed by t h e geometry  n  however,  o f t h e d o m a i n fi i m p l y  the that  g H (Q).  n  e  (cf.[17])  8< '  q } a n d q " a r e i n H^(Q)  $  such  s  n  c  is  ^ ( )  , $ , w i t h Q s u f f i c i e n t l y smooth,  n  "^W.n*  constraints  a real  modes.  2  s  Since  the  e H (Q)  \(i (x)  The r e g u l a r i t y o f $  so  of  of e l l i p t i c  fi  that  (3. 16.2)  is  theory  we h a v e  .  n  regularity  the r e g u l a r i t y in  Vr  1  n  l  smooth,  but  fT (Q);  (cf. [29]),  (3.16.4)  2  a  3.2 The F i n i t e Element  We  now  proceed  approximation first  Approximation  to  of equations  introduce  the  parameter,  partition  fi  vertices  into  = C*^'  x  2j*'  the  finite  ( 3 . 6 ) - ( 3 . 12). Towards  finite  discretization of  describe  element  0  < h  small (J>  29  £  spaces. < 1,  rectangular e  1  2 > 1 -  Let and  this h  -  1,  e n d , we denote  let  elements i  element  1 -  a  be fi  with  g  J  a  -  J.  and s i d e s p a r a l l e l t o the c o o r d i n a t e axes. (i,  i  j),  and  directions,  are  local  respectively.  in  the  h  = diaCQ^),  e  j  sense  that  each v e r t e x k =  in  the  identifiers  The  p a r t i t i o n 25^ = U  there exists  p = sup  For  a constant c  dia(Sph  and  fi  is  e  x^  regular  <r > 0 s u c h t h a t  for  K  the  Sl^)  h  p. Given  positive  e  integers  numbers o f  vertices  t h e number  of  K  and  (or g r i d  elements Q  h = { h  H  In  v  H  equal The  is  (i  1  ••  (Q)  n "l™-  oh = h  s  °  C  e  (3.17.1),  degree  points)  s  v  h\  Xg =  constant,  the  interfaces  its  so  that is  a  This  fact  of  with  <t>Xx );  *k V (x  X  2  }  v  n  e  =  x 1  <  to  is  equal  g  dimensional  x ) 2  to  spaces  h  of  basis are  h) >  =  h  s c  n» •  is  o.  h  polynomials  in  x ^,  The d i m e n s i o n the  lines  set  (  linear  w r i t i n g <p,(x)  ^.1:  x^  of  \ P^\ y  of  r e s t r i c t i o n of  piecewise  by  2^i  x  '  X  = constant 4>^(x) a l o n g  1  ~  k  ~ K  1  ~  of is  and  any  of  polynomial  in  as  product  a  tensor  one  i.e.,  t ( > t>/ ' > i  the  permits 2  <  equal  D  €  H  space  canonical  variable. x 1  r  n> \  s  i n t e r f a c e between elements  1  p  e  = 2 i n our computations).  t o K and  <t> ( )  and N  are defined  ch = c  the  in  is  h  w i t h the p a r t i t i o n  H  where  e >  i n D, , t h e f i n i t e  e  associated  N  1  ~  J  '  1  ~ J ~  30 I  J  <t>.(x. ; 1 lr  where the  =  5.  ,  lr  4>^( ^  a  x  d  n  finite  *j  (3.7.1),  piecewise  [16]  we  analogue  : u. | _ wi  1  H. h  dimension  points.  of  h  The  j * s * J  linear  the  polynomials  in  according  to  Next,  define  the  space  space  W^,  the  introduced  H  of  = 0 Vfi e D, , (2 n T = 0 fi e h e "  in  is  ( 3 . 17. 3)  equal  to  the  number  is defined  boundary  by  = i1'  of  the  l  f  P  i  6 r  [ _ 0, o t h e r w i s e  v  consists  of  h  = \ w , 1 s i s K., w. . € M , v. .(P.) I hi b hi h hi l .  of  (3.17.2)  n  The c a n o n i c a l b a s i s B, o f M,  support  }• , '  = H, ® M, . Oh h  h  B  i s r ± I ,  by  M. = •{ u, e H, h h h  The  ,  js  respectively.  x^,  Pirbnneau  dimensional  = 8.  are  and  and  )  2s  (pXx^)  variables  Glowinski  <f> .(x.  elements  adjacent  1. I '  to  boundary. The  spaces  ^Qh^  a  r  e  m  e  m  D  e  r  the  s  f  a  m  ily  of"  finite  1 00 dimensional  spaces  (Q)(HQ(CI)). properties approximate Al)  If  H.c h  Q^c  V '  We now f o r m u l a t e of  ^(Q),  inf  „ * "h  (cf. [2],  1 + I>ntz0,  llu  -  all  i n the e r r o r  complete and  analysis  in  inverse of  the  [12]).  t h e n Vu  <= H (Q),  £ Ch \\u\\ p  s, 2,  is  some a p p r o x i m a t i o n  which are useful  solution  which  (Q)  fi  €  31  r*0,  r  " ,  r , 2, fi  0<s^  min(r.m)  the  min(1+1-s, r-s) .  where p ^  1 00 A2)  If  i n a d d i t i o n , we a s s u m e t h a t u e W ' (Q.) t h e n llu - Z «  inf  „  -  Chllull,  * "h e  A3)  For  2 s r * 1+1  inf I llu - *ll + hllu - ^11  + hf  Let  k^,  llu - a»  such  n  be a r b i t r a r y  that W ,  ,.  II)  J  + hllu  oo, f2  then  +  _ ; I £ Ch llull 2, oo, fi  numbers  there  exists  r  r,_ 2,  a n d p, q € [l,a>] s u c h a  constant  that  //^ c  C = C(<r, k^.k^, p, q)  e H, ,  h  II  IIv. II. _ h k2,  ^ „,N/q-N/p+kl-k2..  _ ^ Ch  q,  ..  IIv. It, , h kl,p, fi  fi  w i t h W t h e d i m e n s i o n o f t h e d o m a i n fi.  12)  IIv II. _ * C h l l v , II , n l,co,i! n 2 , 2 , fi _ 1  .,  „  ^  2-JV/2. .  n oo,  Let in  H,  h  compute  Q .(x), * n  l  to  the  approximations  ,.  n 2, 2 , fi  , Cx,), ¥ . (x.) a n d <J>fYx,) d e n o t e n  sh  cj.Tx;,  i  -1A-1/N,.  fi  *  ah  finite as  (x),  s  h  Y (x) a  element  linear  and  solutions  combinations  o f H.. T h u s  h  32  $ Cx^,  of  the  approximations  respectively. we  expand  the basis  To such  functions  ^  = pik*k k  *sn  =l*sk*k k  (x)  * an n  (x)  Likewise,  in  (3  * 0h  * 0n H  ( 3 . 13)  the  ^" V  * (x)  = E k  *2h  =l* 2n*k k  ih  (3  •  stream  (x)  n  XJ)  h  functions  '  l  e S„, ,  Ch  , a r e known,  sh  can  3)  18  4)  also  '  (x)  = f, i =1,2 ,  (3  (3.6)-(3.8)  is  defined  ah  = Q°  h  = 0 ,  with C = 1, s u c h t h a t  T h e n f o r n > 0, a s s u m i n g *  18  be  (3.18.5)  element approximation of  G i v e n Q°  hs  2)  - -  (3  follows:  f i n d #,  18  ' "  H  (x)  1 8 1 )  ' -  S  l*k+k k  of  -  as  n  The f i n i t e  =  ( 3  * Cn  (x)  (x)  view  approximated  \n  €  =l*ak*k k  (x)  *h  ( x )  solve  (?",  for  33  ,  ,  i//?^ a n d  ' 18  as  6)  1)  <  +  At  i 2 - f <  8  Q (x)l  3 i ^ '  +  Q* b  n+1  . . n)  ^  J  >  +1  6  (W  (  Q  =*  T  ^  n  U  ^  1  2  >  ^  * V  ^  V ' %  (3.20)  n  uh  h  . that  , ^  ah  +  "oh .  e  tf , .  , V<0 ) + A T *  ah  ^  V  -,n+l jn+1 „ , * , , $> e such ah n On  x  +  = -  (  Q  - Q  l  2  , <t> ), V<p e H^^ h  h  (3.21.1)  '  V  CT  V  ,  K  =  tf*  V  '  b  1  h  ,n+l * i h  B  Finally,  H 2 T - * r  the values derivatives  ah  .n+1 \  we h a v e  V  *h  +  e  (  "  3  cf * h  2  -  1  2  sh  .  (3.21.4)  H .n+1 ,n+l 1 n+1 .n+1 h • ^ h = - n r r r * h % • T  $  +  2  t o compute  o f V^^h  W  e  w o u  the v e l o c i t i e s  ^ ^ <  and  obtain  i n each  (3.12).  velocity  t o H , but they would  34  )  (3.21.3)  o f the stream f u n c t i o n s  would not belong  Oh '  H  .  J  =  +  2  '  1^ a h d f i  =  1  1  layer  (  3  -  2  2  )  from  I f we t o o k t h e vectors  which  2 b e i n L (Q); i n o r d e r  to  force  them  derivatives  to  of 0,  be „,  in  o n t o H,. n  l,Zh  we  H^, For  project  i  = 1,  n+1 ^  2  the  define  n+1 V' lh  dx^  (3.23.2)  •  n+  lh  orthogonally  dXg  2 The o r t h o g o n a l  p r o j e c t i o n f r o m L (SI)  -  (V J in n+  It  remains  vorticity is  used.  V*}* , ih  to  on  describe  the boundary  This  is  in  the  (3.7.2),  we h a v e f o r  is  Be  rectangles exactly This  l  the  h  the  computation  when  the n o - s l i p  = -\  support  integrals  to of  by  by  of  the  relative  boundary  condition  the d i s c r e t e analogue taking  —  =  0  of in  that  J] J-u,ds, +  he  h  adjacent  Thus,  M^.  i = 1,2,  V^^Vu.ds  he  where  space  given  (3.23.3)  the  done by c o n s t r u c t i n g  (3.7.2)  f  h  is  V0. e H , . h oh  = 0,  1  onto  l  of the  h  h  M^.  Since  boundary,  (3.23.1)  yields  35  Vu,  h  e N, h  '  is  Be then  element  (3.24.1)  by  we  composed can  element  of  compute of  Be.  the  ^n+1 _ = ih  where wall is  is  |n| point  , .n+1 .n+1 ^ - ( 0 . , .,- i/»., ) . 2 , ih,W ^ih,I ( |n| ;  to  locally a  context  as  r  along  the normal  the  first  interior point  second  order  formula  Wood's  formula.  triangles  or  yields  an  algebraic  easily  solved  boundary  T  the d i s t a n c e  V  .  3  A.,  n+1 ih,I - —=, 2  curved  by  points  In  the  is,  a more  elements  linear  in  known  as  system  „„  d i r e c t i o n from Formula  I. in  finite  general  set  support of  of  method  general,  not  since  very  difference up;  such  the  as  (3.24.1)  which  large  the  (3.24.2)  Be,  equations  Cholesky  ^  (3.24.2)  can  number  (see  be of  [16]  for  details). By  computing  (3.19)-(3.21) systems of  one  i  W  obtains  [  s Q  S[* ] a n+1  T[* ]  sparse,  ]  the  denotes banded,  entries given  7 = [RJ  that  following  appear  algebraic  in linear  T  1  }  =  [ R  = [R  i2  3  4  a  ( 3 . 2 5 . 1)  ,  1  = [R J  n+1  [  integrals  equations:  S//*  here  the  column  symmetric,  (3.25.2)  '  ]  (3.25.3)  ] ,  (3.25.4)  ,  matrix.  The  positive  matrices  definite  S,  V.  l  matrices  and  T  are  with  by  t . . = f V<t> .V<t> .dQ , u  u  36  ( 3 . 2 6 . 1)  s. . = f CV0 .V©>, . + \ <p, .<t>, .)dQ ,  (3.26.2)  2  V  l i J = \ *hi*hj (  ^hi^hj  +  Q  V  2ij = J / ^ h A j  where v = (—^-AtA ) u  The the  (2,2)  is  unique.  Conjugate  the  quadrature  positive definite,  v  "*hi +hJ V  a n d u = (1 +  c a l c u l a t i o n of  Gauss  +  then the f i n i t e  The  systems  Gradient  (JCG)  (3.25) method.  37  (3.26.4)  '  )da  Ate) .  integrals rule.  (3.26.3)  '  )da  is  Since  performed  are  element s o l u t i o n e x i s t s  and  solved  the  using  matrices  are  all  by  by  the  Jacobi  CHAPTER III  A G A L E R K I N - C H A R A C T E R I S T I C ALGORITHM FOR THE H Y P E R B O L I C STAGE  This Chapter  is  d e v o t e d t o t h e f o r m u l a t i o n and a n a l y s i s  the f i r s t  stage of  the algorithm.  prototype  equation  the  the  solution  curves is  of  that  of  the under  solution  coincides  solution.  Our  weak  main  s o l u t i o n by  methods.  is  An  very  t h i s purpose,  linear advection  which  flow.  For  constant  weak  (transport)  along  interesting  here  is  of  with  the  e n d we c o v e r  equation,  this  the  equation the  weak  classical  approximation  a c o m b i n a t i o n of G a l e r k i n and  Towards t h i s  as  characteristic  assumptions  everywhere  concern  the  property  regularity  almost  we t a k e  of  of  the  Characteristic  t h e c o m p u t a t i o n a l d o m a i n Q,  2 n £  fij  £ R  [Q^  is  supposed t o be a r e c t a n g u l a r  domain),  with  a r e c t a n g u l a r g r i d and a s s i g n one p a r t i c l e  t o e a c h node o f  grid.  the p a r t i c l e s  the that  T h e n we t r a c e b a c k  the p o s i t i o n s  c h a r a c t e r i s t i c curves the magnitude  the p a r t i c l e s , grid points particles  at  X(x^, t  x;  particle  The  +  of  so  x^ at  t)  the values instant  denotes  of  would  be  interval x  t+r;  the values t).  position  t + T is  linear  38  assume  transported variable  by  at  the  p o s s e s s e d by  the  Here and at  and  along  in  the  instant  sequel,  t  of  the  a t x. . k  "representation a  is  the dependent  t +T a r e  the  instant  mathematical  a time  the dependent v a r i a b l e  the p o i n t s X(x^,  that at  idealization  during  of  the  of  combination  this of  physical  Dirac  delta  functions is  c e n t e r e d a t X ( x ^ , t + T; t ) . B u t t h i s  not very  convenient  regularize  it  substitutes  smoother  functions.  which as  piecewise  element  space  t ) . Therefore,  given  with  their  view,  this  of  limited  centroid  centered  achieved  the p a r t i c l e  one  to  extension We  choose  functions  the departure  delta  amounts  velocities.  the basis  at  if  f o r the Dirac  parcels  functions  s)  of  points  solution at instant  = Yp<- (t)<t> (x k k  where  a^t)  denote  s;t).  Since  ^ . C x ~ ^^ k'  '  ^  S  the p a r t i c l e  equivalent  to computing  interpolation  of  ^k^ ^'  =  X  the  X(x^,  s = t + x  the values point  that  t) ^  s;  s)),  at  the points  n  e  n  To d e t e r m i n e  s o l u t i o n onto  the grid  We p r o v e  k  o f w(x,  e l e m e n t s p a c e H^.  we p r o j e c t  t).  S  - X(* >  k  the values x  to the f i n i t e  u(x,  by  of  s o we must  by u(x,  of  is  functions  point  particles  computations,  This  piecewise  physical  are advected  finite  is  a  point  smoother  t+x;  mathematically.  From  replacing  f o r numerical  representation  belongs  the weights  a n d show t h a t of  values  this  w f x , s)  aAt) of  by  cubic  a given  algorithm  XYx^,  is  u^t)  this  is  spline  functional  conservative,  2 unconditionally Moreover,  for  stable  sufficiently  superconvergent  1.  in  the smooth  L -norm functions  and  convergent.  the s o l u t i o n  at the foot of the c h a r a c t e r i s t i c  is  curves.  Preliminaries  Let  the computational  w i t h L i p s c h i t z boundary  f,  domain R  2  39  Q be a n open  subset  i s the two-dimensional  of  R  2  Euclidean  space.  We  now  B-splines  as  I =  and  2  [c,d]  positive  introduce  well  as  of  that  and /  I  a = x  definitions  that  A  <  c = x  For  positive  r,s  spline  functions  I -  11  which  < x  A =  1  {x  2  and  [a,b], be  I, J .}  J  2J  1  are  satisfy  = b II  < x  21  Let  1 2  ix  1 1 1 2  splines  Q = I x I . Let  =  1  respectively,  < x  of  their properties.  the domain  such  &  partitions  some o f  assume  integers  ^  the  <  < x  22  = d . 2J  integers,  we  of order r over  define  the  linear  space  of  as  I  l S  . (I ) = J S(x ) e H (I ):D S(x ) = 0 for x in [x .,x . ]\. r,Ai l l p l l l ii n +i ' r  for i = 1,  , I. Here  = -{ f e (f (^)  H^d^)  r _ 1  f  r  e LPaj  continuous,  2  \ ,  (I ) = L (I ) V  P  i  i  Specifically, degree  r  -  derivatives The is  '• D f is absolutely  2  D  and  r  1  a  which  2  is  up t o o r d e r r  linear  space  defined similarly.  product  spline  splines  S  of  S(x^)  €  continuous -  and  a  with  P°ly  n o m  i l a  continuous  2.  spline  We a r e  1 S  functions  of  order  i n t e r e s t e d i n the c l a s s  . © S . d e f i n e d as r,Ai s,A2  40  follows.  s of  over  1^  tensor  of  S  . ® S . = •{ S(x ,x ) = S (x )S (x ) : S e S . , r.Ai s,A2 ' 1 2 1 1 2 2 1 r,Ai  We n o t e  . 9  s  r, A i w h e r e k = min(r,  the  of  order  S  a result  functions  the notation,  to Q of  the tensor  To  this  spline  e n d l e t S(Q)  be  Furthermore,  of  ^ case  r  a  Typically  (&)  h  is a special  [36] f o r t h e s p l i n e  F o r a n y f e S(Q)  o n e f . e S. , (Q) s u c h t h a t Sf h k,h  ii f, ii s e n f ii , f e L hp p i i i ) Quasi-linearity (f  =  There  Stability.  II C f +f ) . 1 2 h  product  b y S.  approximation  linear  c S(Q)  L (£i)  assume  space  P  that  S  of  for 1  satisfies  properties.  Uniqueness.  ii)  we d e n o t e  a lemma w h i c h  i n which S i s defined.  the f o l l o w i n g  cn; ,  s).  due t o Widlund  S.  p  6  s , A2  p s n, m = 0 , l , . . . k .  i)  c i/ '  . . We now f o r m u l a t e s, A2  procedure  i  .  to simplify  restriction  with  . V  s,A2'  2  that  s  In  S e S  +f ) 1 2  P  there exists  one and o n l y  f.. h  is  a  constant  C,  such  that  en; n sen; . in LP  II s p  || f  (Q). II l h i p  + II f  -f 2  II h  , 2  f p  e  ,f i  2  sen; iv) functions  Optimal  For  accuracy.  a l l  f II f . - f II h p  where k denotes  * Ch |f |  the order  k  p,k t  ,  of the spline.  41  sufficiently  smooth  Lemma  1.1.[36].  i)-iv),  Let the approximation  S  satisfy  then II f.-f II h p  ^ Ch \f\ p,r  (1.1)  .  r  0 < r < k.  for  An ^ {x  procedure  alternate  a n d {x . }  {x ^.>  i  i  l,...,I+r  =  constructive  definition  as f o l l o w s  [7]. Enlarge  involves B-splines  2  . }* 2  and  J  to  non-decreasing  of  S  . r, A i  and  the sequences  sequences  {x , }  I + r  and  , the a d d i t i o n a l points being otherwise a r b i t r a r y . For  B-spline)  and j =  of order r  1, . . . ,J+s ,  t h e i-th B - s p l i n e  (s) f o r t h e s e s e q u e n c e s  is  B. (x) = (x . - x .)[x ., . . . ,x . ](x-x)/\ i i +r  i,r  i i  i i  ii+r  (j-th  l  +  (1.2)  f o r a l l x i n I . S i m i l a r l y f o r B . (y), f o r a l l y i n I . i J>s 2 H e r e [x ^, .. .x^ ^ ] i s t h e r - t h d i v i d e d d i f f e r e n c e o n t h e f u n c t i o n  r~ l (x^-  x)  +  =  max(0, x^- x)  keeping  x  fixed.  From  (1.2). the  following recurrence r e l a t i o n i s obtained.  (x) =  B. 1  ,  X  B. Cx; = i,r  i i ~ " 0 , otherwise.  1  1  — X  x . n+r-i  '  X  X  . -  -  - x . i i  X  ii+i (1.3.1) X  B.  (x) +  i , r - i  —  X  — B . (x). x . - x . 12+1,r-i 12+r n+i (1.3.2)  We r e c a l l thesis  some p r o p e r t i e s o f t h e B - s p l i n e s  (cf.[7]).  42  which a r e needed i n t h e  Bl)  B. i,r  BZ)  F o r any  = 0 f o r x not  (x)  Cx ,x  1 2  i n [x  11  .,x  . ]. 11+r  in Q  )  E^/V^'V " • j=i  I  1  -  (1  J  1=1  4)  B3) S.  ,(Q)  = span  k,h  B4)  Let  iB.  > i,r  *  the  (x  )B .  l  j,s  non-decreasing  (x)\. '.\ l  'i,j=i  2  sequence  {x  .}  I + r  x  {x  111  the  interior  •{B. 1  knots  (x )B . 1  V  X = ix • y  1  • For  J  the s t r i c t l y  of  form  J + S  1  B-splines  increasing  sequence  •)  of data points,  J  the  spline  2 J 1  S(x  l  , x ) = V c . .B. 2  fx ;B. Cx ) 1 j,s 2  i,r  IJ  ,  (1.5)  a g r e e w i t h the g i v e n f u n c t i o n f a t x i f and o n l y  Schoenberg-Whitney s o l u t i o n of  theorem  (1.5)  (cf. [7])  i f and o n l y  B  °  #  e q u i v a l e n t l y , i f and o n l y  '  2J  X  1 1  <  *2J  1  <  43  the  unique  '  V  (  1  -  6  1  if  x . < y . < x . 1 1  guaranties  if  if  l.r<*M> J.s<*zJ>  B  or,  sequence  J  .}  1, J—1  2  x ix  111  will  the corresponding  (x )\-\' ._  JrS  1  of  2  X  J  , V i +  2J S' +  ,  (1.6.2)  r  V J  '  '  )  2.  Description  of the Algorithm  The C o n t i n u o u s  2.1.  Consider  the  Problem  Cauchy  equation f o r u(x,i)  » at  +  .  u  w(x,0)  ^  assume t h a t  denotes  v  w  the  scalar  = o  Dt in  fi  ,  (2.1.1)  ,  (2.1.2)  the v e l o c i t y f i e l d  i s incompressible,  = 0 in Q  t o t h e boundary  =  u.  i.e.,  ,  (2.2)  T o f fi, i . e . ,  0 .  (2.3)  require  u ( x , t ) € L Yo,r;l/ <'fi,>,> . compute  characteristic  the  solution  curves  system o f d i f f e r e n t i a l  of  X{x,s;t) equations  44  (2.4)  ,co  a  To  advection  the material d e r i v a t i v e of w i n the flow  u-n|  We a l s o  =  = U (x) o  V-u  and t a n g e n t  for  i n t h e c y l i n d e r Q^. ,  d  where  problem  (2.1) of  (2.1)  we  introduce  which  satisfy  the the  X(x,s;s) = x  We  state  solution  without of  proof  (2.5)  and  the  (2.5.2)  existence  summarize  some  P r o p o s i t i o n 2 . 1 . Under condition solution is  in  integer s  C° (0,T;W °(Q)).  regularity  (2.4) there  Furthermore,  i,C  of  m  k  a unique  (2.5)  X(x,s;t)  w h e r e X(x,s;t)  -  denotes  f l u i d which i s  n  as  u(X(x,s;x),x)dx  s  the p o s i t i o n  at point x at  We n e e d f u r t h e r  then for a J J a e H , 1  c a n be e x p r e s s e d  = S  for some  m  t  x  that  X(x,s;t)  e C°(0,T; L (Q)).  a  of  CD  the  results.  exists  assume  k £ 1 u is in L (0,T;W ' (Q)),  |a| s Jt, t -> D X(x,s;t)  now  uniqueness  t -» X(x, s;t) of the system (2.5) such that  The s o l u t i o n  of  and  results  of  at  instant  (2.6)  ,  time t of  the p a r t i c l e  s.  the s o l u t i o n  of  (2.5)  which  we  formulate.  Lemma  2.1.  \s-t\  sufficiently  Under assumptions small,  (2. 2),(2. 3) and (2.4) and for  x -» X(x,s;t)  homeomorphism of Q into itself  is a  with Jacobian  quasi-isometric  determinant  equal  to 1 a . e. Moreover,  L~ \x - y | £ \X(x,s;t)  - X(y,s;t)\  1  w h e r e L = exp(\s-t|•|u| 1  n  )  .  i , «>, Q 45  £ L\x - y\ ,  (2.7)  Proof. F o r that and  a l l s,t  there is  is  a  unique.  mapping  in  2. 1  continuous,  as  guaranties  On t h e o t h e r h a n d , follows the  and  its  since  X(x,s;t)  that  inequality  (2.5)  as  (2.6).  The  transformation  is  by  again  continuous.  Liouville's  J(x,s; t) ~ 1 a . e .  X(x,s;t) ^  1,  can  being  is  itself.  of  one  It  is  is  to  J(x,s;t)  guaranties  that  and  II-II  2,fi  proved  (see  a quasi-isometric  is  a  determinant  = 11-11  then to  of  the  above  =  det(DX(x, s; t)).  t  -> J(x,s;t)  condition  (2.2)  is yield  [25],  Th.  mapping  1.1.7)  of  II • II  class _ and  that  simple of  the  matter  to  C '*(Q), ml  II • II  *  ->  m are  m,p,Q  check  transformation  x  is  that 1  if  a.e.  the then  «  condition well  to  quasi-isometric  classical  s o l u t i o n of  (2.1)  is  given  by  (2.8)  o  is  it  obtain  w(x, t) = u (X(x, t;0)) .  It  = x.  2,Q .  The u n i q u e  This  and  inequality  m.p.n Jacobian  a  •  be  It  one  easy  a  determinant  w h i c h maps Q i n t o Q , t h e n o r m s  equivalent.  as  X(x,s;t).  (2.3),  Gronwall's  x -> X(x,s;t)  formula  exists  = X(X(x, s; t), t; s)  applying  as  (cf.[18])  considered  neighborhood  inverse  given  2.1  be  mapping  Jacobian  Proposition  Remark. I t  this  can  the v e l o c i t y s a t i s f i e s  Hence,  homeomorphism.  a  maps fi i n t o  (2.7)  2.1  U(x) f o r w h i c h X(x,s;t)  into  U(x)  well  from P r o p o s i t i o n  s, X(x,s;t)  fixing  from  Proposition  we h a v e  neighborhood  By  ^CxJ  (0,7)  holds  known  under  [29]  very  that  46  for  weak  regularity  some  integer  assumptions. m i  1,  u  e  Z. CO r;l/ ' fn;; e o  n  J  problem (2.8).  p  and u € \T' (^) o P  ( 2 . 1 ) belongs  t o L°°CO, T;]/ ' (Q)) 1  Now, l e t <£(•) € 2)(R ),  J  P  solution  of the  and i s g i v e n  where D ( R ) = \ <p e C°°(R )  2  has compact s u p p o r t  t h e weak  2  2  by :  <p  }• , then  f u(x,t)<p(x)dx = f u (y)^(X(y.O;t))dy R R  ,  (2.9)  J  s i n c e t h e J a c o b i a n d e t e r m i n a n t o f t h e t r a n s f o r m a t i o n x -» X ( x , t i s e q u a l t o 1. a . e . I n ( 2 . 9 ) y = X(x,t;0)  2.2.  .  The D i s c r e t e Problem  L e t W be a p o s i t i v e i n t e g e r ,  At = T/M and t  = mAt f o r 0 s  m s W - l . L e t u^(x, t.) be an a p p r o x i m a t i o n t o u(x,t). that the  u,(x,t) h  satisfies  approximate  interval  conditions(2.2)-(2.4).  trajectories  [t , s] a r e given,  of the points  For s x  T  For  T  The  i n t h e time  the departure  point  w h i c h r e a c h e s t h e p o i n t x a t i n s t a n t s=t  error  e(x)  (2.10)  ,  s At  x = A t , X,(x,s;t ) denotes n m  trajectory  = t , m+i  a c c o r d i n g t o ( 2 . 6 ) , by  X, ( x , s ; s - x , ) = x - T u, CX, f x , s ; s-c), s-c)dc n o h h 0 s  We assume  = X(x, s;S-T)  X,(X S;S-T)  -  L  n  47  m+i  o f the  committed  in  satisfies  Lemma  approximating  the f o l l o w i n g  2.2.  conditions  e(x)  that  (2.2)-(2.4),  Tn  Subtracting  |e(x)| ^ r '  '  T  o  ^ T  and  i  ( e x  u(x,t)  fulfill  m in the  interval  the  error  T 1  any two set  finite of  step  from  1  1  that  we a p p r o x i m a t e First,  this  a) d>, (x) i s  the bound  t h e weak  a particle  is  Towards t h i s  •  s o l u t i o n u(x, t) b y of  projected  as  (2.8)  onto  e n d we i n t r o d u c e  defined  <p (x)  (2.11).  approximation  approximation  functions  k b) supp$ (x)  .  '  inequality yields  t  1,2, .. .K and 1 £ i  with  (2.10)  |e(e)|de  03, Q <  e l e m e n t s p a c e H^.  k  i t follows  (2.11)  u(X,(x,s;s-c),s-c)\dc h  then  cut-off  1) .  Iu, CX, (x, s;s-c), s-e) h h  process.  up,  -  u(X(x,s;s-c),s-c)\dc  Gronwall's  instant  _)  |u, (X, (x, s;s-c), s-c) h h  |Vu|  + Si 0  Applying  (2.6)  P  Q  1  o  H  (2.10)  is bounded by  Proof.  is  u,(x,t) h  then for any integer  03,  a  by  for any real x s u c h t h a t 0 s x ± A t  |e(x)| *  At  trajectories  lemma.  Assume  [0,M-1] and  the  follows.  the  the For  set k  =  l  basis of  £ I, 1 s J < J bilinear  in x  and x  1  = ^  - h .  11-1  _ n l i  _ ' 1  s x  n  ^ ^x  l i  £ h,. l  2  n 2  j_ '  , - h  l i  c) <f>^(x - x^) = <t> (x .), Q^x) k  48  h  1  2 J  .  -''  V i  z x  2j - i e  'J' £ h  2  "i^k^ ^k-i' X  ..  2J c  a  n  o  n  i  c  a  Thus  t i  ~ = <*k H  Notice  that  intervals — Qj^x)  shifted  support  of  4> (x) t o  1 , 2  k  CR ;  is  clear  ,  2  C4/u ;j ^ C x j d x k  the  k  J  <f> (x) e l /  ii)  that  ( 2 . 12. 1) = 1 ,  (2.12.2.)  = meas(supp<p (x)).  is.  IS.  X, Cx, , t ; t ) be h k m+i m  which  the  . ,h .]x[-h . ,h .] , V i, j . I t 11-1 11 2J-1 2j 2 compact s u p p o r t i n R and s a t i s f y  i)  Let  is  suppfy^Cx)  [-h  have  where u  (x)  reaches  consider  the  0. Cx k  <f> (x), w h e r e  is  d>, (x - X, C x , , t k h k m+i make s e n s e  grid  knot  x, k  - X , C x , , t ^ ;t )) h k m+i m S( •)  k  the departure p o i n t *  is  the  ; t )) m  instant  5(x  trajectory  t  .  Let  us  X, C x , , t ^ ;t )) h k m+i m  -  measure. is  the  m+i  =  Dirac  which  at  of  required ^  Another for  our  *  property  of  algorithm  to  the f o l l o w i n g : If  iii) then supp^Cx  -  for  some  k and  k  m+I  supp<p.(x) K  n supp^Or  X (* .t ;t )) h  j  m  r\ supp<j> .(x) * 0 , J  - V  X  j ' W ' V  ;  *  (2.13) Let  us  check  Assume regularity  that At  of  =  i i i )is OCh;,  the f i n i t e  satisfied. by  virtue  of  (2.7),  e l e m e n t p a r t i t i o n one  49  (2.11) has  and  the  0  IX. Or ,.t  h  j  m+i  it  ) - X , ( x , , t : t ) \  m  h  £  Since  supp<t>,(x)  from  (2.14)  grids, that in  not for  k  (3.14) to  iii)  in  (I,  0(hAt  +  m+i  +  \X(x.,t j +  * es i s  uniform,  m  h  ~ X,(x h ) - X(x  |u - u. | . - A t h oo, Q  o f o r d e r 0(h), For  this  there  ;t  m+\  - X (x  m  \ X ( x . , t : t J k m+i m  t  k  k  ,t  an  in  compute  the  ;t  Before  doing  simplification.  solution  Cartesian  components  B. Ax, ) 1,4 lp  and  ^(x^)  a  B. .(x„ j,4 2q t  b y A^and A^ Let " { i ^ - j grid  let  points  at  is  the main  the proposed points  introduce  we s e t  result  index  j  are  X, (x, , h k  t  m+1  t  , respectively. be t h e v a l u e s instant  t^^,  50  \x^pW 2qt' x  1  m  ~  P  Such m a t r i c e s a r e of  the approximate and  A  a  this to  ,; t ). m+1 m notational hk .  J  points  of  algorithm  X,Cx, , h k some  ~ *'  with  matrices B. 1,4 1  .(x.) 1 ~  g  invertible. solution  symmetric  m  )\  )\  clear  ensures  points  (xf, xf, _,), a n d d e n o t e t h e hkl hkz ), g e n e r a t e d b y t h e B - s p l i n e s  the g r i d  at  the  us  Hereafter,  r  to s t a t e  a d e s c r i p t i o n of  so  whose  m  )\  m  . (2. 14)  i n suppcp. (x). k  conditions  weak  element  ;t  ;t  m+\  structured  also  least  m+i  m+i  then i t  condition at  t  j  ;  reasonably  exists  |u-u, | - A t H h e h oo, Q  s e c t i o n and p r o v i d e  the  ,t - , t )  +  satisfied.  . . . ,K)  j  such  We a r e now  £ J,  is  ^ ;t ) f o r a n y y m+i m  a n d Bj  \X(x  t h a t x , e suppd> .(x - X,(x.,t ; t ) ) . Next, k J h j m+i m t h a t y e supp<p (x), t h e n i f one t a k e s y f o r x . i n k J i t f o l l o w s t h a t suppd>, (x-X, (x, , t , ; t )) approximates k h k m+1 m  assume  X,(y,t h  £  m  \x . - x , | + OChAt J k  necessarily  any  zn+i  r\ supp<t> .(x)  that  (1,...,K)  up  k  at  positive  definite  matrix  with entries  a  =  rs below the proof  of  t h e Theorem  (<t> , d> ). The r s  product  (see  2.1)  A[u> ] = [B]  (2.15)  m+1  satisfies of  [B]  the  theorem  represent  the approximate  that  follows.  the values  at  2.1.  instant  s o l u t i o n w, (x, t ) a t  h Theorem  In  (2.15) t  of  the  entries  b,  k  the p r o j e c t i o n of  the departure p o i n t s  xf\ .  m  nk  -JB. Jx,)B. Jx„)\ .' . ,of ' i , 4 1 j,4 2 i , j = l for the space S „ ,(Q). Then at any instant t , Consider  the  set  I  J  {  B-spline  basis  0 s m £ W-2,  \  4, n  the entries  m  of the matrix  [B] are given by  j ^ J / ^ A ^ ^ X ^ ; .  t h e coefficients  Moreover,  Vk i n  c"! . satisfy  the  2,2  K.  (2.16.1)  relation  A ^ A ^ c " ; = A/"w 7 . 3  For  the d e f i n i t i o n of  theorem  .  In  t h e m a t r i x A see below  general,  a d v a n t a g e o u s t o t a k e A= Theorem  2. 1  (2.16.2)  m  for  well  the proof  structured  states  that  at  any  the  it  is  grids  A f o r r e a s o n s t o be e x p l a i n e d  simply  of  below.  instant  t  the m  approximate by  solution  performing  departure Thus, consists  cubic  w  ^Cx,  t^ ^) +  spline  i s updated at  i n t e r p o l a t i o n of  the g r i d wCx,  points  t ) at  the  points the a l g o r i t h m of  t o compute  the f o l l o w i n g  steps:  51  the f i r s t  hyperbolic  stage  Given i)  [u J :  Vm e  efficient of  the  compute  [0, M-l], way o f  tensor  m  performing  product  by s o l v i n g  [c ] this  step  of matrices.  is  (2.16.2).  using  ~  L e t A[w  m  the  The  most  properties  *  ] = R . The  column  * vector R  can be a r r a n g e d  I and  j < j.  as  I s  an I x J  as  Likewise,  m a t r i x C such  an I x J  m a t r i x R = (r ^J,  the column v e c t o r  [c ] m  1 s  i  <  is- arranged  that  CA* = Y . Then,  for  Finally,  I s for  J s J, 1 s  i  solve  < j , A  * Since  l  [  c  solve  ij  ]  *  =  I y  ij  ]  ' 1 ~ J ~ J •  t h e m a t r i c e s A^ a n d A ^ a r e d i a g o n a l l y  dominant,  they  can  be e f f i c i e n t l y  i n v e r t e d by Gauss e l i m i n a t i o n w i t h o u t  pivoting  (cf. [7]).  number  step  The  of  long  operations  at  this  is  O(IxJ). At  ii)  any  instant  t^,  determine  by  sequential  or  binary  s e a r c h t h e i n t e r v a l s w h e r e t h e p o i n t s (X™, ) l i e . i i i ) F o r e a c h p a i r (xf „ x? x, < x? ,s x , , x. _ , hk hp! hg2 lp-1 hpl lp 2q-l X, _ s x „ , e v a l u a t e the B - s p l i n e basis ng2 2q v. = B using iv)  4  i  CX  the r e l a t i o n s For  j = q, q+1,  ),  m p l  f o r i = p,  ( 1 . 3 . 1 ) and q+2,  q+3  p+2,  (1.3.2).  form  p+3 , ~ m d . = y v.c . . .  J  vO  i• =p  Evaluate  52  i ij  p+2,  p+3  J Y, d.B . JX? .) . j t J J' J 4  hc  2  2  vi)  Finally,  .  , . .  to o b t a i n The  u,  k  solve  the  system  (2.15)  in  operations  O(IxJ)  m+1  number  of  long  operations  (multiplications  and  2 divisions) Here,  r  number  denotes of  departure than 49K of  t a k e n by the  operations points  that  of  + 0(K).  the  is  is  iii) of  to  carry  64K + 0(K).  is  — v)  order  standard  Thus,  B-splines  spline  steps  the  This  the  expensive  i n t e r p o l a t i o n procedure;  Proof  or  the s o l u t i o n  is  of Theorem 2. 1  Mas-Gallic  and  approximation approximate  of the  the  [24]  weak  initial  combination of Dirac  solution condition  view  local  o(x, t) 0  is use  bicubic  nature is  of not  smooth.  Raviart  define  w  K  larger  the  standard  to  may  at  which  when t h e d o m a i n  not g l o b a l l y  we  total  slightly  of  the  According  Raviart  the  procedure  the  t h e B - s p l i n e s may o f f e r some a d v a n t a g e s rectangular  is  point  however,  point.  computations  spline  than  per  Hence,  number  from a computational more  + o(r)  spline.  out  bicubic  4r  ^y^  a  as  [32]  particle  follows.  by  and  a  We  linear  measures  k=i for  some s e t  substituting  of. p o i n t s  iy^P^^ (2.17)  into  (2.9)  53  y^  e Q and  P « k  n  a direct calculation  k  e R yields  .  By  K  = I P u ° 6 C x - X (x  u (x,t)  k  Here ^ C x ^ , t ; 0 ; = y In  k  [32, Th.4.1.]  solution  i s proved  t h e convergence  to regularize  product  with  function.  degree  a  have  cut-off been  proposed  we a r e i n t e r e s t e d o f smoothness  conforming  (2.18)  tothe  ( 2 . 9 ) f o r a n y <f> e C CR .). F r o m a p r a c t i c a l  i t i s convenient  however,  of  2  of view  functions  (2.18)  .  0  weak  .  ,t;t))  h  k=\  P  finite  approximation  types  of  i n the literature  and which  cut-off  Next,  at instant  cut-off  (cf. [32]);  functions  are suitable  elements.  t o u(x,t)  (2.18) by a c o n v o l u t i o n  Several  i n using  point  t o work  we  o f low  with  define  C°-  particle  as  t m+i  K  u (x,t  ^ ) = Tp.u..(t  p  where  m+i  w,,Ct )  hk  m  . ^ k h k r n k=i  ;t ) , t ), m+i m m  function  at  k  process move  vertices their  h  k  {x^}, b y p a r c e l s  centroids.  the value  i s a grid  point.  to replacing  the characteristic of  limited  We d e n o t e b y w ,(x, t  o f w (x, t ) which p m+i  (2.19)  m+i  of  value to a  weight  • «  The r e g u l a r i z a t i o n  point  curves  ph  :t ^ )) ,  m+i  as an approximate  .  amounts  k  u  denotes  p,  and x  o f (2.19)  along  - X (x,,t  i s t o be understood  w(X, Cx, , t  n  )8(x  particles,  which  passing  through  the  extension  moving  with  ) theregularized  form  m+i  i s constructed as follows,  K  u> .(x,t  ph r  ^ ) = Y.p,u,.(t  m+i  .^^khkrn k=i  )8{x  54  - Xjx.,t  h  k  ;t  ))  m+i m+1 A  *(pl  1  k  4>.(X))  k  K  k=i We  note  right  Young's  theorem  2 is  away  °  hk  m k  that  each  h  Y  w ,(x, t  ph  m+i  convolution  k  m+i  2  is  )  p o i n t s X^(x,s;t)  point  a r e i n fi a s o n e d e d u c e s  2  1  t h a t w ,(x,t  m+i  xf, w h i c h a r r i v e s  of  . Fig.2  a t v e r t e x x, a t i n s t a n t  k we l o o k  f o r approximating  Since  H..  h  r  d>. (x-X,(x, , t  k  h  t h e n we c a n s e t u ,(x, t  k  '  ;  coincides  ))  m+i m+i  m+i  h  i t is  depicts a  ^  process  at  t  thesolution  ;t  that the  m+i  ) = w, (x, t  ph  &  the regularization  hie  space  by  ' > '  f r o m Lemma 2 . 1 ,  m+i  J  representation  Now,  *  and given  ) e U ' (Q) f o r a n y t  ph  graphical  P^^^  since  —  i n £ ( L (R ) ) . B y t h e d e f i n i t i o n o f <p  clear  2  i n L (R )  operator  2  m+i  m+i  (2.20)  i n the  w i t h <b.(x),  k  ) where  K  % ' m+i (x  and s u c h  t  )  =  Z k k=i  \  Q)  (  x  )  '  X  6  Q  '  (  2  '  2  1  )  that  (w,(x,t ) , <p.) = Cw, (y,t ), 4>.(y - X, f x , , t ;t)) h m+i k h m k h k m+i m m+  , V k, m,  m  (2.22) where  (u, v)  From  (2.22)  algebraic  =  uvdQ . one obtains  linear  K,  step  i n the proof  the entries  the ' weights  satisfy  the  system  A[u  Our n e x t  that  b  of  m + 1  ]  = [B] .  i s t o show t h a t  (2.15)  f o r a n y k,  [B] a r e t h e i n t e r p o l a n t  55  1  values  k £ of a  bicubic of  the tensor  prove of  spline  this  at  product  assertion  generality.  a =  the p o i n t s  x < 1  Let  < x  By  of f u n c t i o n s for  virtue  of  a , b be  the  on r e c t a n g u l a r  t h e one d i m e n s i o n a l t h e end p o i n t s  = b and  i  X^.  properties  g r i d s we  case without  of  the  consider  w  k  r  J  loss  interval I  K(X ) = I " f <(>Ax - X )<f> (x)dx , V k,r', k  can  (2.23;  r  a  where (x h ; +  k  -  x  k  • v  the symbol  Z,  is  either  then by v i r t u e of  derivatives  in  -{x^}-.  generate  polynomials  k+i  h  ]  '  +'XLk  as  - x  (— h.  • )l/> T  k+i  cubic polynomial  uniform  each  one has  possesses the  in  variable,  (2.14)  Such  the  W  [  X  b  first  interval  i.e.  a well  that  for  and  second  [x  collection  ,x  linear  space  with continuous  >  of ^  (X)dx  if  direction  or  structured  order and  ],  is  grid, an  r  continuous  with  breaking  r+i  x  3  derivatives  56  . Now,  any k t h e r e  6  piecewise 4  in  coordinate  r points  e  ) c a n now be w r i t t e n  piecewise  a  smoothly  KCX^J  x  '  ]  r  is  that  x  <t> (x)dx  ^(^^)  progressively  such  K(X  )4> (X)dx  a  grid  h  otherwise.  - x  J  the  k' k  [x  •  K(X,  Clearly,  k  €  0,  Dropping  x  >  cubic  °f up  polynomials  piecewise to  second  cubic  order  at  the  breaking  theorem  points  ( c f . [7] )  -{x^. ,  P  i,x with  the  s p l i n e S(x)  in S  characterize is  obvious  analog  space  of  (2.23),  .Cn). 4, n  have  one  to  the  In  linear  of  two  there  x  hk  r that  dimensional  exists  a  cubic  case  Since  combination  given of  grids  a n y S(x) of  by  e S  tensor  order  hk  the  in  ; = AlJ"]  r  A are  the  d i r e c t i o n A = A. a  Xf,=  then i t f o l l o w s  entries  -X, ), d> (x)). k r  Hence,  ascertain  takes  SCx  where  the  in  hk  we if  Curry-Schoenberg  s u c h t h a t S(X?, ) = K()CT, ). I n  h  S(x) that  S  JQ)  4,  coincides  to  3  r>  case  According  values  the  two  to  ). I t r dimensional S(x  ,  (2.24)  the  inner  which  are  products uniform  (<P (  in  each  , (Q) c a n b e e x p r e s s e d 4, n  products  of  cubic  x  k  as  B-splines,  t h e n f o r a n y k,  k  Taking  hk  (^^^  ,  L  .  ii  L  I=IJ=I  ^"zhk^  ^ ik'  =  X  X  lhk  i,4  J  2k^  a  D  O  V  4  e  a  n  d  u s  2hk  ing  (2.23)  yields  A*® A* [c ] = A[o> ] m  m  2  1  Remark. I f a n o n u n i f o r m g r i d i n e a c h d i r e c t i o n i s u s e d , * A. T h i s time the  implies  step;  however,  algorithm  limitation) several  that  from a p r a c t i c a l  requires  one  the m a t r i x A has  might  well be  reasons f o r doing  t o be  point  structured  tempted so:  57  to  then  recomputed  A  every  of  view,  and  since  grids  (this  may  be  take  A  =  A.  There  a  are  i) S a v i n g s is  i n c o m p u t a t i o n a l t i m e a n d c o r e memory.  calculated  once  and  for  all  at  the  The m a t r i x A  beginning  of  the  the  next  computations.  ii)  The  conservation  section, would  is  be  conserved  stability iii) the  3.  If  better  a>(x,t), a s we s h a l l s e e  achieved. up  In  to  is  condition  uniform  fact,  if  A were  O(At). - O t h e r  and t h e convergence  t h e CFL  grid  of  is  do n o t less  in  used,  properties  u>(x, t) as  the  suffer.  than  1,  i n each d i r e c t i o n or  then A = A  whether  not.  Properties  In  this  algorithm.  section  we  study  Specifically,  some  we show  important  that  it  is  features  of  the  conservative  and  2 unconditionally  stable  i n the L -norm.  Conservation.  Theorem 3 . 1 .  algorithm Proof. L e t  Let  = meas(suppQ^). For 1 s k £ K and m €  (2. 15)-(2. 16) conserves £ ^-J^t • k  k -m+i1 = A/wm+i Then  58  [0,M-l]  Now,  recalling  follows  (1.4),  (2.15)  and  (2.16)  and  with  A  =  A  it  immediately  r-  m+i  m  k  r o i l  k  Stability.  Theorem 3.2.  Algorithm  (2.15)-(2. 16) is unconditionally  stable  L -norm.  Proof. B y v i r t u e o f  Multiplying  by  ( 2 . 2 2 ) we c a n w r i t e  and summing o v e r  k indices  gives  II u.(x, tm+i ^ ; i l s II u.(y,t f , )" h h m )\\ II Zv.wkf 0 ,k Cy - x hk 2  Now,it of  remains  the )  is  to estimate  inequality. the  the  supports  may  write  + 1  To  shifted of  the second  begin  support  <p (x)  and  with, of  0 Cy  term on t h e  we  recall  -  xf  that  Let  <f> (x).  right  £  hand  side  suppip^Cy -  and  denote  SE.  ,) r e s p e c t i v e l y .  Then  we  SE  J  From  )  ( 2 . 1 4 ) we h a v e  that  i f supp<j> . r\ supp<p i s  J supp<pj n supp<f> = supp<j>Xy - X^j)  n  k  where we h a v e  assumed  \u - u, |  that 59  not  empty,  then  k supp<p (y - X™^)  +  k  _  s  0(h). W i t h  O(hbt), this  in the  information  and  taking  overlapping  is finite  into  account  one e a s i l y  that  the  amount  of  the  gets  Hence  ujx.t h  m+i  )\l s a  + CM)  II u.(x,t h  ),  which  m  )\\ ,  (3.2)  and so s t a b i l i t y .  4. E r r o r  Analysis  Let  u (x)  =  m+1  u(x,t  m+i  according  to  (2.8)  satisfies  o> (x)  = u (X (x n  m+1  where in  xf is h  E^,  h  a shorthand  where  macroelement  t  m  E^  v e r t e x x, , c o n s i d e r k  the of  the  this  in  the proof  J  transformation of  ) ) = C^OO  m  f o r X.(x,t h  support  those  n  m+i  of  C4. 1)  •  ;t  <p^,  elements  m  ).  For  that  which  any x  is,  meet  the  at  the  transformation  x -» x - x  By  ;t  notation  is  composed  m+i  k  + X*  k  = x + «  the element  stability,  E, k  we d e n o t e  m k  is  m+1  60  (4.2]  shifted  the s h i f t e d  SE, . L e t u s i n t r o d u c e t h e f u n c t i o n o) (x) k  .  which  by a" , . hk  As  element  by  3  i s d e f i n e d by  w  We  also  w (x)  consider  ( x ; = w (x + a , , ; .  the approximation  i n thef i n i t e  ml  (4.3)  hk  o> * (x) t o t h e s o l u t i o n m  element space  1  . J^^Cx)  i s defined by  (cf.[28])  ( u < x J > , <p.) = ( J"(X? ), <fi. ) , V k . h k h h k m+1  (4.4)  2 Let  IT^ b e t h e o r t h o g o n a l  respect  p r o j e c t i o n from  t o t h e inner product Cn u, ) = Cu, h  Then  (^^(x)  (u, v),  , V* € #  X  with  i.e.,  h  .  (4.5)  may b e e x p r e s s e d a s  «f Cx; = +1  hh  h Next,  L (Q) o n t o  (4.6)  cxf; . h  f o r a n y x i n £ , we c o n s i d e r  theparticle  approximation  t o (J^*^(x) i n tf, , w h i c h we d e f i n e a s  ZT (x) h X  where t h e v a l u e s  The  inner  =u l u At  HAx  hk m k h  - Xll )  hk  1  .  (4.7)  w. , (t ) a r e o b t a i n e d b y t h e r e l a t i o n  products  hk m  on the right  by i n t e g r a l s o f t h e form  61  hand s i d e  of (4.8) a r e given  h ~< ~^ - *^ (y)  (y  •  dy  E  k  By v i r t u e o f  (4.2)  (4  -  we may w r i t e  m  =  y  and c o n s e q u e n t l y  X  the i n t e g r a l s  L'E,whJ V x formulation  (4.10)  formulation  according  write  as  (4.8)  (  l e t us  hand  the  side  recently inner render  to  , V k .  By  as  an  area-weighting  virtue  of  (4.10)  hk  we  , V k .  « ),4> )  +  (4.10)  k  may  (4.11)  m+i, . m+i, . -m+i, . fx) = w f x > - w, fx) .  , . .„« (4. 12)  n  (4.4)  is  shown t h a t of  the  analysed of  described  products  become  viewed  [27].  %  computations is  be  k  formulation  perform  (4.9)  define  v  The  can  ,<p ) = ( (x  %  Finally,  of  a*)*.(x)dx hk'^k  +  k  The  %k  +  the  in  in  inner  [19].  by  several  and  a  products  Morton  approximation  (4.4)  [31]  of  et the  on al  right  quadrature  method the  right  [27]  have  hand  side  rules  the f o r m u l a t i o n c o n d i t i o n a l l y unstable.  to  might  We w i s h  to  2  ascertain,  in  the  w> (x) b y u^* (x) h m+1  X  J  L -norm, at  a nJ y  the  error  instant  t  62  m+i  incurred  . Towards  by  replacing  this  end  we  9)  set  m+1, . m+i, . m+1, . -m+i, . -m+i, , m+i, . w ( x ) - w, (x) = w f x ) - w, f x ) + w. f x ) - w, ( x ; . (4. 13) h h h h  By  the triangle  inequality  „ m+i, , m+i, ... .. m+i, , - m + i , II w ( x ; - u, ( x ; l i s II w ( x ; - w, ( x ; i n h  + II ZT (x)  The e s t i m a t e given  by the f o l l o w i n g  Lemma in  of the f i r s t  4.1.  For  L (0,T;W ' (n)) m  term on t h e r i g h t  side  (4.14)  o f (4.14)  is  lemma.  u(x,t)  in  and using  1 co  .  - uT\x)\\  X  L°°CO, T;!^* ' (Q)),  u(x,t)  1 2  C ° finite  elements  of degree  k, k £ 1  m+i, , - m + i , .„ „ m, . - m , .„ 0) (x) - u, (x)\\ < II u (x) ujx)\\ h h + |wl . |u - u | _ A t + C h l.oo,C? h oo,Q  Proof. S e e [31] . To e s t i m a t e  observe  Let  by v i r t u e  term on t h e r i g h t o f (4.10)  that by t h e t r i a n g l e  -m+i w, h  L  (4.15:  t h e method  side  used  o f (4. 1 4 )  i n [27].  We  inequality  L  the f i r s t  From  _ . k+i,p,Q  hand  m+i„ _ ,, -m+i ~m+i„ „ ~m+i - u II s II w - w II + II u h h h h  us study  inequality.  l l uV  •  t h e second  we c a n a d o p t  k + 1  L  term  on t h e r i g h t  the relationship  63  u  m+i„ II . h side  ,„ (4. 16)  o f t h e above  it  follows  that  sTUx) - Sf*Vx;n *  ii  n so  that  Using  we h a v e  (4.12)  II  iM"; - »l(x  n  h  n  to estimate  the right  ,  *  h hand  and the t r i a n g l e i n e q u a l i t y  side  The  first  term  h  h  u  hk  h  side  +  hk  - v " Y x + a™ ; i l . ( 4 . 1 8 ) hk  + II Ax?; h  on the r i g h t  (4.17).'  «?;»  m  h  of  we o b t a i n  ii uJVxf; - J?rx«™ ; » s II cx?; - < A x +  (4.i7)  hk  o f t h e above  inequality  is  bounded a s  II u (xf) h m  From  (2.14)  - c / Y x + « J ;|| - C|xf - f x + a " ; | _|vw| _ . ( 4 . 1 9 ) hk h h k oo, Q co, Q  and (4.2)  IX? - fx + a" )l = OihLt + |u h hk H e r e a f t e r we a s s u m e t h a t  II (Ax?; h The s e c o n d  - (Ax  u. I A t ) h  |u - u , | £ 0 ( h ) , n  + a ,;il hk m  term on t h e r i g h t  £  C|vu| _ oo, Q hand s i d e  then  (4.20)  (4.19)  becomes  hAt.(4.21)  o f (4.18)  yields  = J (x? - (x + a™ ; ; - D i A F f x ; - ; d e o n hk 0 Q  64  .  ,  (4.22)  where  F (x) = x + af, + Gfxf - (x + a*)) 8 hk h hk  .  a  (4.23)  Thus  ii  AxJ;  - v (x m  a  ;n  m k  2 s  E  f ixjj-  k  E.  x  -  a i [jiVV Cx;;i dedfie m  2  2  k  e  o  k  Since  FQ(X)  image  o f £, b y t h e t r a n s f o r m a t i o n x -> X, Cx, t ; t ,), t h e n we k ' h m+i m  6  may  change  employing  is  +  a  quasi-isometry  the v a r i a b l e s  in  of £^ onto  the second  £  k  , £  being  k  integral  to  the  obtain,  (4.20),  II v OC?) - v (x + «™ ; i l £ C h A t l l W l l , m  m  m  h  hk  By v i r t u e o f t h e a p p r o x i m a t i o n p r o p e r t i e s Sect.  (cf.  Chap.II,  3.2)  II v (X?) m  Finally,  h  - v (x + a" )\\ s CAth \\ u> \\ m  2  hk  (~u>l (x) +1  - co^Cx),  Collecting  n  ,  term on t h e r i g h t  (4.24)  hand  (4.16).  <p ) '= (Z> (y) - *(y). m  k  t h e same a r g u m e n t  ~m+i w, n  m  2,2,Q  we come t o e s t i m a t e t h e s e c o n d  side of the inequality  Using  of  h  a  from the s t a b i l i t y proof  m+i _ -m m - w, II £ (1 + C A t ) II w, - w, II . n h h n  the estimates  \(y  M  (4.21),  65  - X ^ ,  V k  yields  ,. (4.25)  ( 4 . 2 4 ) a n d ( 4 . 2 5 ) we o b t a i n  -m+i u,  m+i„ ^ , , „ . . , „ -m m„ - w, II s (1 + C At)II w , - w , II  h  f  h  i  + C hAtlVwl  2 Now,from  H  (4.14),  m  +  .  + C Ath H w H 2  oo, Q  + (1 + CAt)-{ll u " - u  w°= w°= <j°, t h e n  h  (4.27) and G r o n w a l l ' s  II u  - w  inequality  K(h,At)  following  Theorem  =  exp(CT).  2,2,Q  . ]•  oo,C?  2  oo, Q  (4.27)  substitutions  into  This  _  m,Q  ,  1  (4.28)  result  brings  us  to  the  theorem  4.1.  For  w(x, t )  in  L°°(0, T ; l / ' ( Q ) 2  2  L ° ° ( 0 , 2 \ V ( f 2 ) ) a n d A t = 0(h), algorithm  n  y  1 , c  °(Q)),u  in  (2. 15)-(2. 16) with C°  ,CO  finite  .  give  II w II  1  + hAt|Vw|  where  successive  h  II s K -7-r— -{h At  h  + K hAt|Vcj|  n  II + II w ° - u™ II}- .  m  that  (4.26)  that  2.2.Q oo,Q  1  .  2,2,Q  J  2  K  _  m  3  f A t ; h | II u II  + 1  u  h  Assuming  n  ( 4 . 1 5 ) a n d ( 4 . 2 6 ) we f i n d  _ ™ || <  1  w  n  elements of degree k = 1 c o n v e r g e s  i n t h e L°°(0, T; L (Q)) Z  norm with error 0 ( h ) .  Clearly,  the  computations  estimate mentioned  (4.28) in  [27]  is  suboptimal.  show  that  for  Numerical sufficiently 3  smooth  functions  o t h e r hand,  the error  i n the previous  committed analysis  p a i d t o t h e f a c t t h a t we a r e u s i n g the  values  at the departure points, 66  may b e  0 ( h ).  On t h e  no c o n s i d e r a t i o n h a s b e e n  cubic spline  to interpolate  and i t i s w e l l  known  that  this  might  points.  In  spline  £  order  in  norm.  r  1 a>  At any i n s t a n t  =  +1  where  S  solution  is  I  h  spline  h  operator  =  j  hj  that  and  u(x,t)  i s r s oo,  solution  PS »(X?.) U  h  interpolant  at the points  D, (x)  the f i n i t e  1 £ s is  u> (x) m+X  of  and p i s  hj  the  approximate  i d e n t i f i e d as the 1 - j £ K }  1  space  (4.29)  ,  hj  f r o m t h e s p a c e l/ = {  dimensional  €  /Y^.Then  m+i, . m+i, . m+i, , m+i, . (x) - w, (x) = w (x) - Io) (x) n r  where  Iu d e n o t e s  right  hand  whereas  side  the  of  term  important  the truncation error  A further decomposition  -  is  P S U W . )  ,  o f w. T h e f i r s t  represents the  concern us because  and c o n t r o l s  instance,  (4.30)  pS(Ax? J h hj  -  the interpolant  second  estimation w i l l  1  e n d , we a s s u m e  this  satisfying,  m+1  7 2  the  ^'"(Q))  + Iu> (x)  iV *  by t h e c u b i c  h  (S^CX^J^Xx)  ,  those  we now e s t i m a t e  the approximate  the cubic  prolongation  step  played  algorithm  integers  t  4  as 0 ( h ) a t  m+i  o>* (x) h  u  the r o l e  n  r and s  T; W ' (£})),  as h i g h  Towards  Z m  simply  into  our  L (0,T;U ' (Q))  €  4.  error  to ascertain  i n t h e maximum  u(x,t) r  a pointwise  interpolation  error  L (0,  yield  (4.30)  term on the  the approximation  error  evolutionary  error.  Its  i t accumulates  a t each  time  aspects  of  t h e method  and t h e n u m e r i c a l  as,  for  dissipation.  yields  = p o A x " ! ; - c A x J . ; ; + (u> (x".) m  P  67  - scAx"\;;  p(Sw (xl.) - ScAx?.» hj h hj m  +  Hence  (dropping  t h e s u b i n d e x Q)  . m+i, .  u)  m+i, ^ . m+i, . m+i, >. (x)\ £ u (x) - Iu (x)\ n oo oo  (x) - w,  r  + |cAx°!; J  - a> (X? .) I m  hj  co  + i ( A x ? J - s<Axf.) i + IStAxf J hj The t e r m B l g i v e s  hj  -  SfcAx? J h hj  with bilinear f i n i t e  m+1  co  mX  ,  (4.31)  elements  £ Ch  \a> (x) - Iw (x)\  h j oo . (B1..B4)  |  2  ,  (4.32)  00  by  i n t e r p o l a t i o n theory.  find  (see a l s o  \d (X .) n  -  m  J  In regard  <Ax? J hj  IcAxfh j J  of  i t remains  the splines  |  ca  £ ClVul  -  oo, Q  M u . - u| h  StAxfh .) | < j oo  t o bound  2 . 2 we  B4.  n  .  At)  oo, Q  (4.33)  Lemma 1 . 1 , i . e . ,  Ch |w| 's,  From  S  oo  .  (4.34)  thes t a b i l i t y  property  one g e t s  iscAxf j - S C A X J J I hj  From  Lemma  [2],[28])  A bound f o r B3 i s o b t a i n e d b y u s i n g  Finally,  t o B2, from  (4.30)-(4.35)  h  hj  £  ci<Ax; - <Ax;i h  co  and assuming  application of Gronwall's  that  inequality  68  w°=  .  (4.35)  co  co°, i t f o l l o w s o n  l w  m i_  m+ij  +  o- = min(2,s)  where (4.36)  K-^fh " At 0  n  s  oo,Q  and K  a  + |vw|  hAt)  ,  n  oo,Q  positive  (4.36)  constant.  The  result  i s f o r m u l a t e d as a theorem.  T h e o r e m 4 . 2 . For u(x,t)  e L (0, T; W ' °°(Q) n W (n)), a  2  L °(0,T;V ' '(Q)), s an integer C  1  a  min(2,s);  algorithm  a(x,t) e  S,m  such that 1 £ s £ 4 a n d cr =  (2.15)-(2. 16)  with  C° bilinear  finite  grid and A t = 0(h) converges  elements on a rectangular  in the  maximum norm as given by (4.36). Note gives  a  that super  framework, estimate regions will  at of  if u  is  s u f f i c i e n t l y smooth,  convergent  result,  in  the departure points Lemma  1.1  is  local  the  s a y s=4, finite  . Moreover,  ,  we e x p e c t  element since the  that  in  order.  Remark. T h e o r e m 4 . 2 may e x p l a i n some o f t h e n u m e r i c a l m e n t i o n e d b y M o r t o n e t a l . [27] u s i n g L a g r a n g e - G a l e r k i n C° f i n i t e  linear to  elements  a d v e c t i o n problem.  to  integrate  In t h i s  simple  the case  one  i s equivalent  they  at the foot  of the c h a r a c t e r i s t i c curves.  u^= u ,  then  yields  £ ;-norm o f  and e r r o r  0(h*/At).  69  methods  were  able  b u t as  to i n t e r p o l a t i o n by cubic  splines  (4.34)  results  dimensional  compute e x a c t l y t h e i n t e g r a l s o f t h e i n n e r p r o d u c t s ;  T h e o r e m .2.1 s h o w s t h i s  2  those  w h e r e u(x, t) i s s m o o t h e n o u g h t h e i n t e r p o l a t i o n a t X?^.  be o f h i g h  with  (4.34)  estimate  I f one t a k e s  i n t h e L^CO,T:  5.  Numerical  Experiments  To i l l u s t r a t e t h e c o n s e r v a t i o n a n d s t a b i l i t y p r o p e r t i e s our  scheme  analysis  as  well  as  we c o n s i d e r  advection of parameters  a cone  of  this  to  verify  is  in a fixed rotating velocity field  Q.  The  are  r  )  f  a p o s i t i v e parameter, X = (X - X ) 1  use  2.9at  steps the  uniform  h = (1/63),  with to  an  the  to  complete  initial  shows  of a  The  (5.2)  1  grid  on  maximum  the  domain  velocity.  revolution  height  activity is the  2  is  [-1,1]x[-1,1],  at  the  observed  at  time e v o l u t i o n of  The  equal  c o n d i t i o n whereas F i g . 3 b  revolutions. wiggle  point  and  H = 100,  -15h, r = 8h a n d t h e CFL c o n d i t i o n e q u a l  =  Q  (5.1)  )  , fix ) .  2  square  X  r  + X  0  u = (-Six We  error the  /  first  the  is  ( 1  The  of  one  " * ' * " L 0 , otherwise ,  where r  validity  two m o d e l p r o b l e m s .  example  r.o; = {-I o(x.O)  the  of  to  the  96.  displays  vertex  is  base  of  F i g . 3a  time shows  the cone a f t e r 6 now  of  the cone e v e r y  number  the 48  87  and  cone.  some  Table  time steps  1  (1/2 2  of and  a  revolution) maximum  values  have  maximum  and  divided  by  for  and  the  beginning  been  in  terms  minimum divided  minimum  to  decrease  mass Tw  values. by  values  H. We o b s e r v e  dissipative  of  their at  that  effects, as  The  square  mass  and  initial  different there  is  they  are  time passes.  70  ,  mass  square  values, instants  mass  For  of  of  while have  conservation.  strong  at  example,  the after  , mass the been As very 1/4  of  a r e v o l u t i o n the r a t e of  time  step  6-th  revolution.  the  is  6*10  v e r t e x of  to  our  decreasing  5  We a l s o  the cone  error  dissipation  observe  is  analysis  to  OflO  3*10  at  the  the  error  in  of  mass  end  of  per the  committed  time step.  0(h/At)  is  square  5  that  ) per  which  of  This  this  at  conforms  case  since  u(x, t) € V ' . 1  Our  second  'slotted* to  test  idea our  0 0  test  high  order this  algorithm [28]  of  the  complete fact  a  the  high  H  We  is  initial  to  Lip ',  where  observe  in  4.  consider  the  by  Zalesak  used  to is  the  figures  control This  is  the  in  the i n i t i a l  the  wiggles  a remarkable  The a b i l i t y o f  under  due  Fig.4a and  the  steps  to  emphasize 1 , p  ,  but  the does  space.  Figs.4b,4c,4d respectively.  ability  of  the  c o n d i t i o n and  keep  generated  achievement  at  of  a  designed  to  t h e scheme  to keep the  2 i t s _L - s t a b i l i t y .  71  The  =4.2  l/  of  schemes. [28].  (See[33]).  outstanding  The  and  time  We  1979  [37]  (m,p) L i p s c h i t z  of  specifically  to  of  96. not  in  CFL  number  in  performance  1 and 6 r e v o l u t i o n s  discontinuities. is  The  is  m,P  a  control  those  the  algorithms.  difference  equal  Lip -  under  not  are  condition  t o m a i n t a i n the shape  is  reported  now  scheme  discontinuities.  compare  c a n a p p l y Lemma 1.1  these  strict  transport  finite  =  show t h e c y l i n d e r a f t e r 1/8,  which  one.  c o n d i t i o n , where h = 0.01,  revolution  we s t i l l  to  results  cylinder  1  is  experiment  this  However,  We  severe  corrected  order  that  belong  more  experiment  this  shows t h e i n i t i a l of  flux  with  using  parameters  height  a  c y l i n d e r p r o b l e m w h i c h was p r o p o s e d b y Z a l e s a k  behind  Munz  is  handle  Table  2  the scheme strong  wiggles  shows  the  time  evolution  of  the  r e v o l u t i o n experiment. are  ,in  some  experiment. square  of  cylinder  mass  progressively  exists  is  similar  time  time  at  the  passes.  to  those  initial For  steps  of  drawn from t h i s  mass c o n s e r v a t i o n .  strong  as  24  The c o n c l u s i o n s  respects,  There  every  of  a  results  the  cone  The d i s s i p a t i o n stages  instance,  to  6  of  decrease  the  rate  of  -4 dissipation  is  10  the  5  after  further  upper  but  few  after  Large  and time  that  asymptotically  a f t e r the f i r s t  second  decrease.  external,  indicate  10  to  revolution. wiggles  lower, steps  their  revolution,  appear  r i n g s of they  are  maximum  + 10% .  72  The  trend  to decrease is  towards  concentrated around  the c y l i n d e r and damped  and  out  minimum  and  the  the values  to a the  slot;  results tend  N°of time steps 0 48 96 144 192 240 288 336 384 432 480 528 576  Conservation mass  Conservation s q u a r e mass  .66972175E+04 .99999999E+00 .10000000E+01  .33569349E+06 . 99706674E+00 .99512210E+00 .99338941E+00 .99178720E+00 .99027655E+00 .98883330E+00 .98744122E+00 .98608897E+00 .98476850E+00 .98347391E+00 .98220088E+00 .98094615E+00  99999999E+OO  .99999999E+00 .99999961E+00 . 10000007E+01 . 10000021E+01 . 10000032E+01 . 10000037E+01 .10000035E+01 .10000026E+01 .10000010E+01  Maximum  .10000000E+03 .92601255E+00 .91213264E+00 .90342497E+00 .89622887E+00 .88989364E+00 . 88427672E+00 .87930044E+00 .87488915E+00 .87096729E+00 .86746379E+00 .86431485E+00 .86144648E+00  Minimum  -  -  -. -. -. -. -.  -. -. -.  0 956667E- 2 101286E- 1 109947E- 1 113813E- 1 115438E- 1 118749E- 1 120397E- 1 120812E- 1 120317E- 1 119158E- 1 117518E- 1 118635E- 1  T A B L E 1. T i m e e v o l u t i o n o f t h e c o n e  N°of time steps 0 24 48 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576  Conservation mass .24120000E+04 .99999843E+00 .10000013E+01 .10000034E+01 .10000019E+01 .99999732E+00 .99999179E+00 .99999869E+00 .99999837E+01 .99999827E+01 .99999836E+01 .99999872E+01 .99999224E+01 .99999870E+01 .10000062E+01 .10000145E+01 .10000233E+01 .10000322E+01 .10000410E+01 .10000497E+01 . 10000579E+01 .10000655E+01 .10000725E+01 .10000788E+01-  Conservation s q u a r e mass  Maximum  .96480000E+06 .40000000E+01 .995827685+00 .11428758E+01 .94983986E+00 .11180027E+01 .94457594E+00 .11307321E+01 .11386835E+01 .94061257E+00 .93735256E+00 .11387771E+01 .93453740E+00 .11348827E+01 .93203288E+00 .11289023E+01 .92976022E+00 .11239973E+01 .92766902E+00 . .11269107E+01 .11285815E+01 .92572481E+00 .92390286E+00 .11293093E+01 .92218474E+00 .11293036E+01 .92055625E+00 .11287153E+01 .11276561E+01 .91900618E+00 .11262104E+01 .91752550E+00 .11244438E+01 .91610680E+00 .11224079E+01 .91474390E+00 .11201438E+01 .91343160E+00 .91216546E+00 .11176853E+01 .11150600E+01 .91094165E+00 .11167015E+01 .90975685E+00 .11185765E+01 .90860816E+00 .11291797E+01 .90749300E+00  TABLE 2. Time e v o l u t i o n o f t h e s l o t t e d c y l i n d e r .  73  Minimum  0  - 150388E+0  --  131663E+0 171569E+0 - 189159E+0 -. 1 9 5 7 6 2 E + 0 -. 1 9 7 6 2 6 E + 0 -. 1 9 6 1 7 1 E + 0 -. 1 9 3 0 1 0 E + 0 -. 1 8 8 8 2 8 E + 0 -. 1 8 4 0 1 7 E + 0 -. 1 7 8 8 1 5 E + 0 -. 1 7 3 3 7 3 E + 0 -. 1 6 7 7 9 3 E + 0 -. 1 6 3 3 2 9 E + 0 -. 1 6 0 6 6 9 E + 0 -. 1 5 7 7 5 7 E + ) -. 1 5 4 6 5 1 E + 0 -. 1 5 1 3 9 6 E + 0 -. 148027E+0 -. 1 4 4 5 7 3 E + 0 -. 141044E+0 -. 137493E+0 -. 133902E+0  F i g . 2 R e g u l a r i z a t i o n of the P a r t i c l e  74  Points  3a In °nd  c  7S  on fo,  the  C  One  F i g , 3b The Cone a f t e r 6 R e v o l u t i o n s  76  F i g . 4a I n i t i a l C o n d i t i o n f o r t h e C y l i n d e r  77  Fig.  4b The C y l i n d e r a f t e r 1/8  78  of a  Revolution  F i g . 4c The C y l i n d e r a f t e r 1  79  Revolution  F i g . 4d The C y l i n d e r a f t e r 6 R e v o l u t i o n s  80  CHAPTER IV  ERROR ANALYSIS OF THE POTENTIAL VORTICITY AND BAROCLINIC-BAROTROPIC MODE EQUATIONS  The o b j e c t o f of  the  second  potential  t h i s chapter  stage  of  the  vorticities.  approximations  of  is  to develop an e r r o r  algorithm  We  also  used  for  estimate  the stream f u n c t i o n s  the  and  analysis  computing  the  error  the  in  the v e l o c i t i e s .  In  2 both cases of  the e r r o r a n a l y s i s  the e r r o r  namely,  error,  then  employed  elements  in  boundary the  the  we  approximation  prove  that  vorticities, to  our  case),  approximation  error  the estimates  element  methods  of [6]  condition.  vorticity are  used.  of  The  ),  with no-slip the  order  given  in  [6]  and  error by  for  of  of  (linear  finite  free-slip conditions  These  results  linear  relative  t h e o r d e r 0(h)  )  when  improvement 81  an  no-slip in  w i t h the  estimate  the  for  the  method  finite  vorticity  proved that the a p p r o x i m a t i o n e r r o r of  two  the  when  0(h).  -  thinks  evolutionary  boundary  [14]  function  [14] g i v e s 1/2  o r d e r . 0(h  0(h  of  the r e l a t i v e v o r t i c i t y i s boundary  optimal,  stream  the  space 2  is  the is  in  one  composed  contributed  variables  is  and  If  approximation  is  are used;  improve  In  the  error  the  which  discretize  conditions  formulation.  i n the L -norm.  c o m m i t t e d i n the a p p r o x i m a t i o n as  parts,  potential  is  the  boundary estimates  for  free-slip relative  conditions of  the  approximation  error  is  Characteristic  method,  equations  the  in  circumvents respect  the  to  component  the  suspected  that  products.  Chap.8].  caused  of  by  error,  our  a  second  Galerkin  terms  the  of  the  variables,  those  terms.  With  method  leads  to  a  (by r e a s o n s  to  be  time f i n i t e - d i f f e r e n c e component  which  is  of  variable  0(h)  methods error  term i s  the  first  compute  analysis  the  on  the  The  error  analysis of  of  methodology of  stage.  It  is  i n h e r i t e d b y many G a l e r k i n  w h i c h do n o t  based  analysis  in  to  potential  given  the e l l i p t i c  the f i n i t e  exactly  element  in  the  -  inner  vorticity  [10]  and  equations  [35,  follows  literature.  Preliminaries  We which  introduce are  vorticity us  the  is  the standard  the n o n - l i n e a r  conventional is  the  c o n t r i b u t e d by the p a r t i c l e a p p r o x i m a t i o n  this  Our  equations  of  there  of  Characteristic  1.  that  is  evolution  that  o p t i m a l a n d more a c c u r a t e  than  and  fact  derivative  evolutionary  However,  o r d e r 0(h)  the  lumping  material  which i s  methods.  by  to  difficulties  the  seen below)  due  needed  in  equations.  consider  u  some  the In  error  order  the p a r a b o l i c  to  definitions  analysis  of  simplify  our  and the  results potential  exposition  let  equation  2 - V u = F i n QKIO.T] ,  t  ul  preliminary  r  u(x,  = 0)  0  Vt  ,  = u (x) n  (1.1) .  82  We d e f i n e  the R i t z  projection with respect  CVPjU,  In  fact,  solution  as t h e o r t h o g o n a l  to the inner product  Vx) = (Vu, VX)  PjU i s  HQh  p r o j e c t i o n P^ o n t o  , ^X e H  the f i n i t e  so that  .  Qh  element  of- t h e corresponding  (Vu, Vv),  (1.2)  approximation  elliptic  problem  of the  whose  exact  2 solution  i s u . T h e L - p r o j e c t i o n IT^ o n t o  projection with respect ClT u, h  Some  results  x)  = (u,  t o the inner product  concerning  established  i n the appendix  w h i c h we t h i n k  so that  w),  (1.3)  the approximation  P ,  pertinent  we i n t r o d u c e  (u,  X) , Vx e Hh .  projection  Next,  i s the orthogonal  to  our  of this  properties  error  of the  analysis,  are  chapter.  the "discrete  Laplacian"  o f a s a n o p e r a t o r f r o m H, o n t o  o p e r a t o r A^,  itself,  by  n  C A 0 , x)  = ~(V<JJ, Vx), tf>, X e H  h  .  QH  (1.4.1)  Note- t h a t A,P,  h  For,  with x €  f A ^ u ,  The u,  h  x)  = ILV  h  2  .  Green's formula  = -(VPf,  semidiscrete : [o,T]  1  (1.4.2)  yields,  Vx) = - ( V u , Vx) = (V u, )  equation  > H ,, takes  2  X  o f (1.1) which t h e form  oh  83  = CTT^u,  consists  of  x) •  finding  ( u  or,  hf  x  )  since  W  r  -  x  =  )  ( U  h ' F  x  =  F  h  = "  X  h  T h e c o m p l e t e d i s c r e t e s o l u t i o n if  +7  :  *  e  •  - 5  that  m  + k V s.(kL.)F.(t l h h i=l  1  n  functions  uniformly  numbers  which  + x .k) , n= 0,1,2,.., I  1 . 1 . The  order  and only  r(X)  ii)  (1.6)  and - l s . C X H a r e i 1 m  1  1  on t h e e i g e n v a l u e s  -{ ^y T  m  a  r  e  distinct  of real  1].  Definition p if  a r e bounded  i n k a n d h, a n d where  i n [0,  = e  discretization  (1.6)  is  accurate  of  if  + 0(X )  A  and for  time  as A —> 0 ,  P+1  0 s 1 < p  m  .  1  t  j  E T s a ; = -4iyCe - E Aj- ) * 0(X ) as X -> 0 . A  m  i=l  To  X  P l  j=0  i l l u s t a t e t h e meaning  J  -  o f r(X),  m = 1,  x  1  and t h e r e l a t i o n s  =  i)  1/2 , r(X) ' ' v  a n d ii)  =  yield  84  and  sAX)  an example w i t h t h e C r a n k - N i c h o l s o n  n  (1.5.2)  of (1.1) i s defined by  1  nu  = rCkhJlf h  rational  i)  '  n  w h e r e w i t h k t h e t i m e s t e p a n d t = n k , r(X) ^ n  kA^,  0h  H  are a l l i n H, , Oh  0  T —> H such On  1  v  >  ( 1  )  Q  if  x)  h-  = (u (x),  u (x,0)  1  (F  t h e f a c t o r s on t h e l e f t  V " V h  (f  = h'  )  scheme.  For this  \ Y , s(X) 1 - A/2 ' +  we s h a l l  2  f o r A —> 0  =  give  scheme  1  1 - A/2 '  1 + A/2 A 1 - A/2 = e 1  = A (e  = A  1  2(2 - A/2;  -  X  2  i - A ; + OCA;  (e -  2  1) + O C A )  x  = 2A C e - 2 - A - A / 2 ; + 0(1) 3  1  4(1 - A / 2 ; observe  latter  n  1  1 - A/2  Also,  ,*3. + 0(X )  that  inequalities  A  2  |rCA;i  <  imply  the  |sCA;i  1,  <  1,  unconditional  for  any  A.  The  stability  of  the  scheme. Let  us  now  introduce  the f u n c t i o n s  v ^, 0 £ 1 £ p,  defined  by  1!  1 T r  v cA; =  i  A  ,(  r  „a  4  )  >  J  p>  It  follows  is  accurate  vAX)  Definition accurate  from of  order  P  1.2.  p if  -  X  p-1  J  1.1  that  the  discretization  (1.6)  if  ) as X —> 0, 1= 0,1,.., p  n  -  J  1  1  of order p^, p  Remark. C r a n k  "  > - I. r.s (X) , 1= 0 1=1  and o n l y  The time  v^X)  ^  p  Definition  = 0(X  o  X\ J  L  -Z-Ar  discretization  (1.6) is  strictly  £ p if  = 0 . 1=0,.. -,P -1 0  Nicholson  and  85  backward  Euler  schemes  are  accurate  and s t r i c t l y  accurate  of order  p  =  2  and p  =  1,  respectvely.  2.  Error Analysis  In  the error  potential the  of the Potential Vorticity  vorticities  upper  layer,  solutions subindex  analysis  is  of the approximate  ( C h a p . I I . 3.20)  since  essentially  t h e same.  i from t h e equation,  solutions  of the  we r e s t r i c t o u r s e l v e s  the analysis  are r e f e r r e d t o t h e upper  Equations  of  the  lower  Hereafter  understanding  that  to  layer  we d r o p  the  the variables  layer. 2 1/2  Let vector  us d e f i n e  £ = (1 + | u | )  , where u i s  a n d |o| i s t h e E u c l i d e a n norm.  n u m b e r s f r o m t h e i n t e r v a l [0, 2n] c o s a . = u ./£ , j = 1, 2. W i t h J  of  q  c a n be w r i t t e n  d i r e c t i o n T = (cosa^,  ,  cosa.^,  dq ^ „  c  Recall  L e t oc^, <x^,  such this  be r e a l  t h a t c o s a ^ = 1/^ , notation the total  J  derivative  Dq  the v e l o c i t y  from  Chapter  the d e r i v a t i v e  i n the  Thus  cosa^).  dq .  II  as  ,. dq(X(x,t;t),  an equivalent  t)  ,„ ..  d e f i n i t i o n of the total  derivative,  Dq Dt In the  terms upper  of  =  _dq_ ds v  ( x ( X t t ; s ) t v  '  s  ;| ' 's=t  (2.2)  (2.1) the equation of the p o t e n t i a l v o r t i c i t y i n  layer is:  86  g  dq(X( g  *'  t : s )  q l  '  r  q  b  q(x, 0)  The  =  s )  A V q(X(x,t;s),s)  , Vt  (2.3.1)  g=t  ,  (2.3.2)  = f .  (2.3.3)  corresponding  (suppressing  + F(X(x, t; s) \ ,  2  H  semidiscrete  version  of  from the n o t a t i o n the dependence  (2.3)  is  o n X^(x, t;s) f o r  brevity):  W  VSTQ*l  *n >  +  --  F  (2  = 0 ,  r  Q*(x, 0)  Q  Q  (2.4.3)  = f ,  1 denotes  ''r "  |u^|  15  (2.4.2)  * here  4  %n 2 1/2 )  a  d  with  From C h a p t e r canonical  n  t h e f u n c t i o n Q - Q', Q' e H (Q) s u c h  II  F  h  Vh  =  u^ being ((3.18.1))  basis of H, n  ?  +  h ^ f- W '  V  that  °  Vt +  the s e m i d i s c r e t e the expresion  a p p r o x i m a t i o n o f u. *n o f Q i n terms o f the  is  uh Q* (x) = I Q* \(*> k n  k  Also, for  from  Q is  Chapter  (dropping  II  (3.20)  the  the superindex  87  • Vt  complete  n  e T .  discrete  * for brevity):  equation  Q (x) - Q(X") n+1  (  R  V  Z>  (F  K  0 ;  ,  T  where y^, y ^  H  r  e  parameters  Q(X^) such  (QCx,  2  h  V  f r o m /'O, 1] s u c h t h a t  t)  V  =  (2.5)  from Chapter  = Q(x,  , Q(^ ) ),  +  '  &H  We f u r t h e r r e c a l l  is  H  ' ^h Oh a  Ux)  A CJ(vlQ n  +  III,  S e c t i o n 2,  £ y^, and  that  = I Q 4> (x>  n  k  k  that  t ; , <(> ) = (<?"(>;, 0 C y n  k  -  k  t  n  +  2  ;  r ^ J ) , Vk  Hence  A[Q] = [B] , and  from  Theorem  2.1  of Chapter  are g i v e n by c u b i c s p l i n e  III  ,  the e n t r i e s  i n t e r p o l a t i o n o f A[Q ]. n  Lemma 2 . 1 . Assuming that  bounded at any instant  scheme (2.5) is unconditionally  stable  2 in the L -norm.  Proof. T a k i n g i n ( 2 . 5 ) jn+1 <P = y Q (x) + V Q(X^) n +  h  2  2  yields  + A )\V(? Q + n+1  H  1  o f [B]  7 Q(X^ )\\ £ \\F \\ \\<t> 2  2  n+K  2  h  88  t, the  Now,  (Q ,  since  Q(X?)) < l/2(\\Q  n+1  II  n+1  + IIQII ; a n d t h e  2  second  2  h term on t h e r i g h t  IIQ  n + I  H  2  hand s i d e  -  HQH  is positive,  < 2AtCIIF  2  n + K  llCllQ  we f i n d  n + I  H  that  + IIQIi; .  Hence  IIQ  H  n + 1  s  s where  ||Q|| + 2 A t l l F ' | l n +  (l + C A t ; i l Q H  the s t a b i l i t y  We n e x t  which  scheme.  theorem  of  ,  C  (2.6)  Chapter  III  has  the accuracy  we r e s t r i c t  ourselves  To t h e a u t h o r ' s scheme  to  the  knowledge  of  used  to  t h e scheme  the case  classical this  i s analysed  C h a r a c t e r i s t i c methods.  been  to  •  t o prove  corresponds  Crank-Nicholson -  n +  inequality.  wish  simplicity,  1/2,  + 2Atllf ' ll  n  obtain the l a s t  For  C  is  that  y^  = n^  =  Crank-Nicholson  the f i r s t  i n the context  We p r o v e  (2.5).  f o r At  time  of  that  Galerkin  = 0(h),  our  2 method g i v e s  an e v o l u t i o n a r y e r r o r  of order  OCAt  + h)  instead  ) a s one would expect from t h e C r a n k - N i c h o l s o n  scheme.  2 o f OCAt On  the other  easy  to  Euler scheme while our  hand,  following  establish  scheme  is  that  of  the steps  the time  order  of  accuracy  0(M+h).  In  this  our proof of  the  sense,  it  is  backward the  latter  i s o p t i m a l when a p p l i e d i n c o n j u n c t i o n w i t h o u r m e t h o d , the Crank-Nicholson  main  accurate  theorem than  the  shows,  scheme  is  suboptimal.  the Crank-Nicholson  backward  Euler  steps.  89  scheme  However,  scheme for  as  is  more  larger  time  In  our  analysis  we  need  L a p l a c i a n operator which i s  Lemma  2.2  The  a  property  discrete  Laplacian  positive  inner product  space with respect  definite  )  x £ ^Oh ^  Let  e  = -(Vij>, Vx)  X  s  u  c  in  discrete  lemma.  -A^  operator  operator  is  a  considered  2 to the L -inner  Proof. R e c a l l t h e d e f i n i t i o n o f A, , h h  the  formulated i n the next  self-adjoint,  (A iP,  of  as  product.  i.e.,  , <P, X e  HQh  that  n  then  x)  (A ip, h  so  =  i f A^i/( = 0 , By  Vx)  -CV0,  virtue  = (*p, Lhx)  t h e n ip = 0 . of  Lemma  = (ip,  yp)  exists  a  =  H0II  2  ,  m there  2.2  set  of  eigenfuctions  of  functions  KO "{XJ^J  w  n  i  c  are  n  the  orthonormal  -A^,  with  the a s s o c i a t e d s e t i^j} of e i g e n v a l u e s . The f u n c t i o n s •{Xj} c o n s t i t u t e a n o r t h o n o r m a l b a s i s o f /7_, . T h u s , a n y V, e Hn, i s expressed  Oh  as  h =  V  Assume  that  G(X)  I h'  k  V*k  (V  is  an  h  *  arbitrary  function  2 spectrum of  -A^  . F o r v e L (Q) we  G(Lh)  = I G(-X j  Oh  )(v, J  90  set  )x,  ,  Xj  J  J  defined  on  the  then  IGCA. ; v l l s n and by t h e P a r s e v a l ' s  IIGCA, ; i l llvll , h  relation  \G(h,)\\ = max\G(-\ .) I h j j For  example,  I I I I = max\-X .\ .  The s c h e m e  Q where  the  (2.5),  with y  = r  boundary  1  /  2  c  a  n  b  written  e  + ksCkA, ;F. (X? ), h h h  = r(kL.)Q(X?) h h  n+1  = 2  (2.7)  +1/2  conditions  have  already  as  been  imposed,  so  n+1 is  Q  i n HQ^*  I + -^-A — -  r(kA.) = r  1  S  '*V  =  Lemma 2 . 3 .  A t d e n o t e d b y k,  ~ —  *  2 +  =  A A  and  -  J  h  .  £—+ , 2 s J s K_ ,  4"* • 2  J  " - -k-  >  J  r  defined  l / i t h P^  2+2 s 2, the following  l  s  J *  which  '  9 )  1 s s s  ,  (2.10)  .Sf & f \L  X  where, for our problem, the constant C = KRe , constant  ( 2  holds. S  X  o .  S  q - q\\ + h\\V(P q - q)\\ £ Ch \\q\\  IP  K  by (1.2) and q € H (Q),  inequality  (2.8)  is independent  number.  91  with K  of h, a n d Re is the  another Reynolds  S e e A p p e n d i x C h a p t e r IV  Proof.  We now with  proceed  respect  solution  the  rate  Q(x,t).  If  the  smooth  of  by  in  the  of  of  main  in  time  that  to  error,  of  is  the  type  vorticity.  which  no-slip  boundary  t h e n " the  the is  is  the  error  free-slip  approximation  error  2  )  error  fact  that  the  regularity  respect  conditions  with in  with  is  0(h).  no—slip  this  case  conditions introduces  regardless  the  solve  of  type  This  of  error  92  term  boundary finite  while elements  in  to  shows  linear  finite  loss  and  potential  2 L —norm,  Chapter  boundary  t h e e q u a t i o n o f q(x, t).  by  conditions  belongs of  an  the  linear  boundary  q(x,t)  approximation  to  and  the  produced  time  of  space,depends  that  with  a  perpetrated  prove  we  boundary  convergence  in  rate  error  for  0(h  approximation  in  the g l o b a l  is  conditions  conditions  Thus,  conditions elements  of  q(x,t)  quadratic  of  chapter  approximate  (see  t h e method u s e d t o d i s c r e t i z e the s o l u t i o n  on  this  the  space  follows)  q(x,t)  of  vorticity  and  A further analysis  approximation  result  convergence  potential  2.1  Q(x,t)  space.  the  function  Theorem  convergence  that  state  to  reasonably  linear  to  •  the is  rate  due  to  (Q.) a c c o r d i n g II.  The of  conditions  with the of the to  particle  order employed  0(h) to  Suppose  Theorem 2 . 1 . i)  €  q(x,t)  ii)  q  e  t  .... m )  2  S  3 d q .2...1. ^ - e L (H ) , ,3 dx  r  denotes  the  derivative  along  the  characteristic  Then  II? -  0  1  1  , C^h*{ \%n  ^  2  where  }  S  +  C  and C^^ are directly free—slip  bounday  + ^  Si2tQ  "^Th (H )  for  1 £ s £ 2, Vt e 10, T]  a>  2 2,„s. „2 d q .2,.2. e L (H ), V ^ - e L (L ), . 2 dx  T  curves.  1  S  L (H )  dq —r— "  where  p| W ' (n),  ff (T2;  boundary  conditions  02  k 2  t  +  2  h  C  (  0 3  H  H  S  according  and  to Chp.II  L  ^ L , Q  proportional  conditions  + llgll»  )  =  1  (3.16.1)  S  •  T  to Re, s  (H )  (  +  -  2  and s  for  U  )  = 2  no—slip  and  (3.16.2),  respectively.  Proof.  Q  Set  n  - q(x,  t ) = (Q  n  - P q(x,  t ))  2  = e" +  P  "  + ( P gC"x,  t ) - q(x,  .  t^)  (2.12)  B y Lemma 2 . 3  l l p H £ Ch \\q(x, n  It  remains  S  t n n  = C h { llg ll S  s 2 Q  to estimate 9  n  0  e H„,  +  llg " 2  . We f i r s t  t  L  note  r / /  s  ;  }  that  .  (2.13)  the R i t z  Oh p r o j e c t i o n o p e r a t o r P ^ g (x, satisfies  t> = ^(x, w  t h e f o l l o w i n g Lemma. 93  t) €  , for g  e  ,  Lemma 2 . 4 .  Let P^ be the Ritz  projection  operator  defined  in  (1.2). Then  M*  a an  p eJS= e L_ 2 dr dx q  Let  Proof.  .  q  be  Sx  a  differential  +  \^\ ) .  of  arc  along  .  •  the  characteristics.  |5x|  = LU1  2  1/2  Then «  / ,. q 1 liaxi^o  »  L  P q(X)  virtue  interval  of  the  Lemma,  n  =  = \h (  F  p  ^-af +  \  ^  " Wi* ( P  q*  = q - q',  q'  2  S  each  rds  *  p  (\(-.t ),s)\ ;S  s=t  <I<X <*.t:s).s)\  1)}  e L (H )  in  ^ =  ~ 4s  1  = F. (X, (x, t;s),s)\ h h s=t where  satisfies  t)  equation  n+1  *  W /  w, (x,  ,/ t h e s e m i d i s c r e t e  It , t  ST)  dP q* = ?—-w-—  h  ' Wh +  - 5x)  above  3w  '  -  A t  2  £  (X  1  1 5 x 1  - P q(X rjr-,  2  By  T C T  M  *  P q (X) - P q (X - 8r)  1 im i' |Sx|-»(A At->0 lim  I  - q  (X)  e  h  sst  t * s,t £ t , , n n+1 such  that  q' \  = q  b  (2.14)  Vt  €  10, T]  Remark. N o t e t h a t A, w, makes s e n s e b e c a u s e w, (X, ) i s d e f i n e d b y h h h h -  94  wAX?) h h  (VP^,  The is  , w h e r e b y P,q*(X?) I n  = P,q*(X?) 1 h V)  solution given,  1  /  1  +  = (Vq*(fy,  X  V) X  we d e n o t e  , X e H  Qh  l / o f the corresponding  , x£ € Q .  totally  1  the s o l u t i o n o f  discrete  scheme  according t o (2.7), by  =  2  -  +  (  2  -  1  5  )  w h e r e £ , , = r ( k A , ,) a n d kh h S..F.Cxf; kh h h  = sfkA, h  We f u r t h e r d e c o m p o s e  9  = (Q  n  n  - l/ )  h  CX. (x, t h n+1  +k/2),t+k/2) n  n  9 as n  + (W" - wjx,  1  t  h  Here  = z" +  t )) n  R  n  .  (2. 16)  II -  (2.17)  x z  "  +  = ^u2<x?) kh h  3  +  u s . ( i khh h  - P ;|^ , 2 3T 7  hence  i+l„ ^ IIZ" II +I  *  ,r-  h,;/«ni. .  „  /  T  h  t h e scheme  i s stable,  also,  we a s s u m e  that  theorem o f Chapter IZCx";il  n  pieces  I  hk  Since  All  _ o »fl<7*  ii £nzcxJJ;ii + kis^i ie ma - ry|f n~  together  then  |£ | i s b o u n d e d .  I I I we h a v e *  | £ . . l < 1 a n d |S | < 2 ; kh kh  (1 + CJOIIZ !! . 71  give  95  From  the stability  \\7 \\  -  n+1  where  C  l i z " l l £ CkWT^W + kC'WCI  and  inequality  C  are  - Pjp-  1  constants.  II , OT  By  virtue  of  Gronwall's  a n d Lemmae 2 . 3 a n d 2 . 4  IIZ""*" !! £ C h | l l q l l 1  +  S  0  w h e r e C = KReexp(K'T),  1 |  -4  K a n d K*  _ L L T  L Cff ; 2  }  S  '  are constants  (2.18)  independent  of h  and A t . The m o s t d i f f i c u l t estimate of R. n  n  +  =  1  R  n  +  1  V  Note  part of the proof  that R ^  t  - w,(x, h  t .) n+1  X  + (E.,w,(X?) kh h h  satisfies  n+  - w,( , h  .) n+1  i s concerned w i t h the  = E , , C W C X " ; - w, e x " ; ; + kh h h h + kS.jP.Z, pkh 1 h ST  - A A,P.q*)(X, ) H h 1 h n  u  (Al, To e s t i m a t e A l we n o t e  that  \\Ai\\ £ \\w(x?) - wjx?)\\ h h h  \E,,\ kh  < 1 ,  £ iii/cx"; - W\/X";II h h h  term on the second  i i i / r x " ; - w.(x?)\i h h h according the  last  proof  to  £ (i  the s t a b i l i t y  term on the r i g h t  given  in  Chapter  A2, A3)  (2.19)  .  (2.20)  so  +  llw.fx"; - w.fx";il h h h h  The f i r s t  i n e q u a l i t y i n (2.20)  + CJOM/  1  theorem hand III  96  - wjx, h of  side to  .  t ;n , n  Chapter of  i s bounded by  (2.20)  evaluate  III.  (2.21)  To  bound  we m i m i c (4.16).  the  Thus,  identifying  functions  Chapter  III  u,  (4.16)  Present  (x)  J^Ax) w,  (x)  >  w,  =>  w  >  h^^h^  from  '  v  h we h a v e  Analysis  h h (4.16) and (4.17) o f Chapter  » V C X " ; - wjX?)\\ h h h h U  III  that  £ UwjX?) - w*(X?)\\ + \\w*(X?) - v.(X?)\\ . h h h h h h h h (2.22)  From  (4.7)-(4.10)  of Chapter  V Next,  we n o t e  Chapter  III  that  takes  III  it  >  (w =  +  follows  V > * • v  the function  v(x)  defined  the  following  expression  t  ) - a(x,  t^)  by  in  (4.11)  the  in  present  situation:  v  Using  n + 1  (x)  = q(x,  the procedure  n + 1  Pl  described  in  Chapter  III  a n d Lemma  2.3  yields  llw. (X?) - v.(X?)\\ £ CJik\Vq\ h h h h 2 Thus,  from  (2.20)  and (2.22)  \\W(X?) - wjX?)\\ h h n + C-Wc|v*g|  2  oo,  Qj.  +  _ + C.h k\\q(x, t ; i l _ „ Q 3 n s, 2,u S  oo,  T  t h e term A l i s bounded by  £ (1 + C.kH]/ 1  1  - wjx, t )\\ + n n  t )\\ , V[t , t ]. n s, 2, Q n n+1  CjfkWqCx,  3  97  (2.23)  The t e r m A2 may b e w r i t t e n a s  Taylor expansions  Y(X(x,k;k),k)  2 k  along the c h a r a c t e r i s t i c curves  = Y(X(x,k;0),0)  + k-^Y(X(x,k;  2  Jc  Y(x(x,k;s),s)\_  d  ds  +J  ^0  =Q  s), s) |  s  =  Q  +  2 2 /  K  ycxrx,ic;0;,e;de  ds  give  *  2  = " l{  "  P  "V  +  tn+2 et — ^ , tn '  T  ;  T  Likewise,  - d T  t o e s t i m a t e A 3 we e x p a n d  Characteristic  curves  tn  U.  n  's-0  A  .3 ^ q CXJx.t dx  r  J  k  ;T),T)  i n Taylor  2^  .2  's=0  .  (2.24)  ds  >> dr\ J  series  along the  and o b t a i n  2  1  tn+2  dx  h  3  2 ( t  +  n 4" - ^V/172 +  98  i N  } •  (2  -  25)  Then,  A2 + A3 =  2  k  j  - l-jr j=0 J  d  * <%) + * J + tfj +  J  ,  q  ,  (2.26)  ds  J  where  and  ,  i/kA^P*  , stand f o r the i n t e g r a l remainders  TJ^,  i n (2.24)  (2.25). Let  wjj d e n o t e  Pf^—.  <J(^>  • The f u n c t i o n s  KkA  ds  J  A)w , J  J  admit a s p e c t r a l r e p r e s e n t a t i o n i n terms o f t h e e i g e n f u n c t i o n s of  kA^A^. I n d o i n g  so,  the factors  terms o f t h e e i g e n v a l u e s  o f k A^A^ b y  lj(kA^A^)  are given  in  1 (X) = 1 - r(X) + s(X) = 0 Q  = 1 - s(X) + (X/2)s(X) = 0  1JX) 1AX)  2  so  W  '  that  = 1  -  s(X) = v  the f i r s t  '  X  1- X '  term  on  the r i g h t  hand  side  of  (2.26)  becomes  j=0 where  J  ds  J  i s a p o s i t i v e constant  We now t u r n  less  than  our a t t e n t i o n towards  1.  t h e terms  3?  n  „ „.  Note  that  tn+2 + IIK^H s e y e  3*  \\p - U L ||ds .  ^ tn Also  99  ds  (2.27.2)  llft"ll £  In  view  that  of  c  6  \  k  (2.23)  l l i i  in  and  n+1  A  2 ds  P  "  (2,27.1.2.3)  |£,, | < 1 because  \\R \\ = l l l /  H h 3  t h e scheme  is  into  account  we h a v e  5  n  2  .2 *  V^l-V  + C k sap II t£s£t , n+1 4  taking  stable,  ^ _ ,s,„ ,  1  -  (2.27.3)  - wjx, t JII - III/ - wjx, t )\\ £ h n+1 h n  n + 2  h  '  S  and  C.kW]/ - wjx, t )\\ + C.hk\Vq\ _ 2  d  ^ oo,Q  + C„h kllqCx, 3  An+1  ^  r  + C  5  k  ds  ^tn  < l^-iL* 2  1  . ...  t ;il n s, 2,  n  ^ „  +  , 3 ds  II V  H*,^"' ^  Hh  1,2 ds  1  (2.28)  d a* %-\\ , 2 ds 2  It  remains  (1.4.2)  t o bound  the terms  ,2 *  h  To e s t i m a t e  is  d a* a n d IIP, 5-11. 2 , 3 ds 3  From  we h a v e  IIA P  P^o  IIA.P, h 2  in  (cf.[38])  IIP  \\P  J  -±4_|| 1 , 2 ds  q  ds  then  3  3 * ii *  ds By G r o n w a l l ' s  d  3  II we n o t e t h a t  there exists  q  2d  3 #  H  D  _ ,2 *  £ IIV -4-ll . ,2 ds  by  3 * 3 * cnypI- -^n s C I I V - ^ I I .  ds  i n e q u a l i t y (2.28)  100  3  Friedrich's  C independent  d  ds  d *  IIVP^-^-^H £ I I V — - ^ l l ds ds  the d i s c r e t e  a constant  *  becomes  .  Since  inequality  of h such  that  s C,h t llgll.co. <>. + C_h llq ll , + 6 n L (H ) T 0 s,2,Q.  - w.(x, t h n+1  1 t  n  S  S  n  +  IIV  ds  ds'  n  c m \v \ 8  n  q  (2.29)  Hence,  putting  yields  together  the estimate  (2.13),  (2.16),  (2.18)  and  (2.29)  (2.11).  Remarks. Notice  i) the  that  derivatives  flow.  For  less  rapid  of  along so  The  along  last  curves  the  the evolutionary  ). F r o m  in  evolution of  error  of  this then  point the  and  time e r r o r  remark  ii)  one  is  the  curves  of  than  view,  time of  of in  of the  q  is  the  t  time  more a c c u r a t e w i t h l a r g e r  variables this  elements, if  that time  101  the  steps  one  along  term  At  gets  the  belongs than  which i s  takes  scheme  particle  rather  =  to  given ui  0(h ),  closer  to  steps.  the backward E u l e r  now Oik),  deduces  from  thereby,  Crank-Nicholson  iii) A similar analysis the  norms  larger  the approximation  of  "optimality" with larger  that  of  the f i n i t e  its  1,  of  the  variation  curves  comes  the f l o w ;  s  £  the  the use  (2.29)  1/2  m  flows  permits  the approximation e r r o r of  s  are  characteristic  characteristic  term  characteristic  0(h  the  2  accuracy.  to  by  multiplying k  dominated  such  approximation  to  terms  t h e scheme  l o s s of  ii)  q  convection  direction, without  the  which  is  scheme  optimal.  the C r a n k - N i c h o l s o n steps.  shows  From scheme  this is  Our  iv) methods inner  analysis  for  the  products  proposed  shows  that  Galerkin-Characteristic  convection-diffusion  are  not  by Morton  et  exactly al  equation  calculated,  [27]  and Benque  as  et  in in  al  which the  [4]  the  methods  ,  lead  to  2 an  error  estimate  appearing  in  regardless  of  might  the  the  norm w i t h a  evolutionary  the degree  of  term of  component  the f i n i t e  of  order the  elements used.  0(h)  error, Thus  one  t h e a p p r o x i m a t e G a l e r k i n - C h a r a c t e r i s t i c methods 2 methods i n t h e L - norm.  Error Analysis  In  this  second  of  t h e B a r o t r o p i c and B a r o c l i n i c Modes  section  barotropic,  $,  order  we  elliptic  II).  of  the approximation of  Our  a d e t a i l e d account  their  respectively.  CV  2  analysis  of  completeness and  the  equations  Chapter  For  study  approximation  a n d b a r o c l i n i c , * , modes.  of  4>  L -  classify  a s 0(h)  3.  in  is  elliptic  based  on  problems  we w r i t e h e r e element  of  B o t h modes s a t i s f y  (recall equations  i t c a n be f o u n d  finite  error  the  by f i n i t e  the equations  to  (1.7)-(1.9)  standard  in Ciarlet  the  theory  elements;  [12]. satisfied  by  approximations  Thus  - A ;* 2  s  = 0 ,  in  Q_ , T  102  (3. 1)  (V  - A j)*  2  =  2  a  Q  - q  l  2  , in Q  T  (3.2) *  |_ = 0, a I  V $  = b ,  2  Vt  in  ,  Q  r  (3.3) * l  = 0 , Vt  r  ,  where H  l l q  H\  -  B  22  H  +  q  ~  +H  2  '  F  (  Cx, t ; + c a ; *  *fx, t; = *  fx; ,  a a n d C(t)  is  V  where ¥ ,  sh  For  rv^,  C V  instant  = (b ,  V  where *  h  ah  Note element  A C*  £  2  +  %n> V  x2(  with c -  ch  ;  %  +  € S  any  II  a h  4  )  (3.5)  (1.6).  element approximations  ™sh> V  (  "  s  d e t e r m i n e d b y Chap.  The f i n i t e  3  =  0  satisfy  ' *h V  *0h '  €  <f> )  =  h  4> ), h  -  6 )  1.  e !P ( h e r e a f t e r we d r o p  ,  ( 3  %  -<  Q  -  l  e H  4> ),  Q ,  h  2  0 h  the superindex  V0  h  £  //  ,  0 h  ,  n),  (3.7)  (3.8)  a n d $ € /7_, . Oh that  0^  -  Q  approximations  £  in  (3.6)  of  -  103  and in  b  ft  in  (3.2)  (3.8) and  b  are  finite  in  (3.3),  respectively.  Due  to  this  fact  some  preliminary  required before proceeding with the standard L e t <p(h) a n d <p b e t h e s o l u t i o n s  2 V <p(h) + c<p(h) = & ,  work  is  analysis.  of the e l l i p t i c  problems  i n fi  h  (3. 9 . i : <p(h)\  = 0 .  T  2 \7 <p + c<p = 6 , i n fi (3.9.2) <p\ = 0 , r  where c i s a p o s i t i v e c o n s t a n t , such  that  element  9^  —» 9  weakly  approximation  to  in  1 1 9. e H, c H (Q), 9 e H (Q) a n d h h 2 L (Q). L e t be t h e f i n i t e  <p(h) i n  Since  fi  is  a  convex 2  polygonal  domain,  t h e n b y r e g u l a r i t y t h e o r y (p(h) , cp € H (Q) f]  and  Hg(Q)  * 2.2.a 9n  where C  are constants  Lemma 3 . 2 .  Proof.  From  £  c  2  2 -V (<p - <p(h))  "  -  independent  <p —> cp strongly (3.9.1)  l l e  (  3  -  1  0  -  o f h.  in H^(Q).  and (3.9.2)  one o b t a i n s  + c(<p - <p(h)) = 0 - 9  h  i n fi , (3.11)  <p - <p(h)\  = 0  104  2  )  Since the  9 ^ —> 6 w e a k l y i n L (Q), <p(h) —> <p w e a k l y i n H (£2) a n d b y  usual  compactness  argument  lim \l<p - <p(h)\\  Lemma 3.2. (3.9.2),  Let <p(h) and &p be the solutions Let <p^ e  respectively.  approximation  "  m  = 0.  be the finite  to <p(h). Then the following  *  ~ ^i,2,n  - V  c , l e  V"J  where C = C(Q) is a positive  of (3.9.1) and  ^  (  h  )  inequality  element holds  •  ~ ^i,2,n  <3  12)  constant.  Proof. L e t u. , </>, € H . , t h e n  h  (v(  h  Uh n  - <p ), v u ;  9h  h  h  c(<p - <t> , u ) =  +  h  h  h  ( 3 . 13)  (V(<p(h) - <t> ), Vp ) + c(<p(h) - <t> , p ) . h  Taking  h  h  = <p^ - <f>^ a n d u s i n g  Friedrich's  h  inequality  (3.13)  yields  ^h-^i,2,Q^ i^ K  We n e x t  ( 3  write  " Vl,2,fi From  -^i,2,n-  (h)  *  I*  -  *< ti,2,Q h)  (3.14)  105  +  **  (h)  ~  Vl,2,fi  -  1 4 )  "  - *hh,2.a  *  ^  ~  *  ( h ) l  i.2,a  +  2  K  i  n  f  ^  (  h  )  *h 1.2.a  -  l  *n On eH  (3.15) It  remains  (3.15). on  t o bound  Note  (3.11)  that  the f i r s t  term on t h e r i g h t  by v i r t u e o f t h e Lax-Milgram  hand  side of  theorem  aplied  one o b t a i n s  " • ^ l  2 , 0  l  1  4  j  f  3  ,  f  l  -  f  l  h - U D l  From t h e d e f i n i t i o n o f t h e d u a l norm by a n a p p l i c a t i o n o f Schwarz  ( s e e Chap.  and F r i e d r i c h ' s  II)  following  i n e q u a l i t i e s one  gets  '*  " *  By s u b s t i t u t i n g We  ( h ) i  *V  1.2.a  (3.16)  into  finite  element  "V •  (3.15) y i e l d s  a r e now i n c o n d i t i o n s  standard  9  to apply  (3  (3.12).  theory  16)  •  t h e arguments  approximation  -  for  of the elliptic  problems.  Theorem 3 . 1 .  Suppose  i)  i)  Conditions  ii)  Regularity  - iii)  of Theorem  conditions  2.1  (3.15.3)  hold.  and  (3.15.4)  of  Chapter  hold. Then  for any instant  "*a " *ah"  J  *  C  / {  *1 Q  t e T n  ~ 2Q  ti  +  106  W*l  ~  V'-.Q. }  +  *" °" ' T  II  II* - *  II £ Ch i Z  ™ s - * s h * *  where  are  dependent higher  lib, II + | v f g , - q.)\ t\ 1 Z  2  n  y  03,  _  i + H.O.T. J  / W 3 / 2 . 2 . T '  C  positive  (  constants  independent  on Re, g = 1 on the boundary order  of h but  T and H.O.T.  3  -  1  ?  )  directly  stands  for  terms.  2 and * i n t h e L -norm  Proof. We f i r s t e s t i m a t e t h e e r r o r f o r * a using  d u a l i t y ideas.  method  (see C i a r l e t  doing, I*  S p e c i f i c a l l y , we e m p l o y t h e A u b i n - N i t s c h e  we h a v e - *  a  [ 1 2 ] , Theorems  t h a t a t any i n s t a n t  .11 £ ah  3 . 2 . 4 and 3 . 2 . 5 ) .  In  t  tf,hll* - * JI, . _ , 1 a ah 1,2, fi  (3.18.1)  II* - * J I £ W _ h l l * - $ j | , h 2 h 1,2,0.  where  are constants  From "°"l  Lemma  2 fi *  Chap.  II,  '*a  '  n  3.2,  HQ(^)>  Sect.  *ah"  so  (3.18.2)  i n d e p e n d e n t o f h. the  equivalence  the approximation  between  I°Ij 2  properties  of  fi  a n 0  "  H^ ( S e e  3 . 2 ) a n d T h e o r e m 2.1 we g e t  *  l  C  h  2  {  "l Q  ~ <V  +  { " h"  | V  ^1  b II ++ IvCg, - q )\ II* - #|| £ CJi \ lib. | V h 2 I h 2 ^ 1 1" 2  0  -*2 o3,Q  }  )[  T  oo,Q n  T  °-  +  } + + H.O.T '  2 The  estimate  of  in  t h e L -norm  is  given  a p p l i c a t i o n o f T h e o r e m 8 . 7 i n Oden a n d R e d d y  107  by  a  [ 2 9 ] . Thus,  direct  3/2,  Now, and  f r o m Theorem  (3.22))  3.1  i t follows  2,  Q  and  '  the  inmediatly  relations  (Chap.  II  (3.11)  that  (3.19)  To  estimate  (3.12),  the  velocities  (3.23.2))  and  Theorem  3. 1  barotropic  the  this  relations  chapter  to  (Chap.  that  II  get  (3.20)  0(h).  l  states  use  (3.16) of  llu. - U.ll = l  we  the  approximate  a n d b a r o c l i n i c modes c o n v e r g e  solutions  of  quadratically  the  in  the  2 L —norm is  to  the  obtained  are  for  linear  velocity  exact  combinations  components the  reasonable  to  expect  fact,  4. N u m e r i a c l  Experiments  In  order  first  to v e r i f y  the  they  shows  is  stream  that  will the  of  functions  modes.  functions,  that  the v e l o c i t y  same d e g r e e  expressed  stream  (3.20) of  of  are  of  In  The  the approximate  derivative  rate.  solution.  in  because  However, terms  then  it  converge rate  convergence  of is  they  since the  the first  numerically  with  a  lower  of  convergence  error  estimates  of  order.  the v a l i d i t y of our  108  for  the  vorticity  results the  of  and  the  stream  some n u m e r i c a l  purpose  situation  of  of  such a  illustrate  function,  examples.  experiments  baroclinic  our  Hopefully,  with  computations  oceanographical We  have  boundary  to  be  calculated II).  case  higher In  The  is  as  The  our  experiments,  of  the  is  the  domain  r o t a t i o n a n d number numerical  homogeneous c a s e . an  exact  solution  Under  of  feature  these  (OJ,I/»)  solution  is  no-slip  is  because  test  the  is,  . *'r *n'r  = 0  109  •  estimate  (1000  square  the it  do  Note not  is  easy  2 2 sin —^— ,  for  order  that  km from  introduce  nonrotating  numerically  UX  (see  a  2  nx  have  of  with V 0 = w ,  2 l —^—  they  to a space  layers to  The  procedure.  density.  conditions  to  ip(x,t) = A(t)sin  =  thesis.  the  one  error  solution. The e x a c t  the  realistic  unknown,  our  fi  our  additional  with  are  because  no r o t a t i o n a n d c o n s t a n t  any  of  to  i n the f u t u r e .  numerical  xlOOOkm),with analysis,  wish  more  first  O(Reh) no m a t t e r i f u b e l o n g s 1.  'real'  this  in  method  a  only  in  be c a r r i e d o u t  the boundary  one  than  we  the that  c o m p u t a t i o n a l l y more d i f f i c u l t .  par  second  simulate  method  the  here  emphasize  analysis  two r e a s o n s .  condition is  to  instead  this  test  the v o r t i c i t y at  Chapter this  to  condition for  valuesof  not  proposed  situations will  decided  t h i s boundary  is  theoretical method  present  We s h o u l d  ocean,  Galerkin-Characteristic  we  to  and  choose  calculated  _ 2  27rx.  7ix_  0  2nx_  o)(x,t) = — — A ( t ) [ c o s — ^ — sin —j— + cos—j—  _ nx.  sin —^—J ,  u(x, t ) \ = u , r  where  A(t) = A(1 -  b  exp(-t/T)).  B y d i r e c t s u b s t i t u t i o n o f w a n d iji i n t o t h e e q u a t i o n  =  D W  Dt  A.A  +  H  F  o n e c o m p u t e t h e f o r c i n g f u n c t i o n F(x, t). We h a v e h,  run several  Reynolds  number  At C F L = max|u|—r—. for  Re, a n d C o u r a n t — F r i e d r i c h — L e w y  Tables  .  h  experiments w i t h d i f f e r e n t values  3,4,5 give ~,  the experiments,  decreased always h  2  i n order  constant.  takes  t h e v a l u e s o f ln— —~ ~  "  f  ~ h " f  j l f | |  A(t) = 0.64A, w i t h A t c o n s t a n t f o r  except  those  t o make  with h = 2  the s o l u t i o n  Consequently,  the values  A linear in  condition  t h e v o r t i c i t y a n d s t r e a m f u n c t i o n i n e a c h e x p e r i m e n t . The  e x p e r i m e n t s were r u n u n t i l all  of  50, w h e r e  converge.  T a b l e s 3 , 4, 5 s h o w s  m a x | u | was  t h e CFL c o n d i t i o n i n c r e a s e s  20, 25, .. . ,40 a n d d e c r e a s e s  regression  i t was  of the values  as h  2  as  = 50.  of the v o r t i c i t y error  that  II w - o> II L  IT—!T  N  = A. + B .hRe , 1 = 3, 4, 5 ,  Hull  where  4  3 0  -3.96, B  c  =  -2.75,  B  3 0  1  = 0.95, A,,  4  = -3.19, B-. = 0.87, A  4  5  c  = 0.80 .  According equal  1  t o Theorem  t o 1; h o w e v e r ,  B^ a n d B^. T h i s  2. 1 t h e c o e f f i c i e n t s  B  should  be  there exists discrepancy i n the values of  might  be due t o t h e s m a l l  110  number  of  points  =  used i n f i t t i n g the s t r a i g h t  l i n e i n b o t h c a s e s . The v a l u e  w h i c h was a d j u s t e d w i t h a number o f  By than  the  one  used  a  linear  Similarly, stream  function  in  B^  and  regression  error  shows  ,  B^ for  that  points  is  very  the  the  these  twice closed  values  of  larger to  1.  of  the  are  proportional  the  vorticity  to  2 h Re,  with  Figures  5a  and  5b  show  the  in  exact  and  the  case.  approximate  r e s p e c t i v e l y of the s t r e a m f u n c t i o n i n the r u n Re =  solutions, 1000,  s i m i l a r d i s c r e p a n c i e s as  h * = 50 At = 2.30h.  F i g u r e 5c i s the p o i n t w i s e e r r o r  of  the s t r e a m f u n c t i o n . F i g u r e s 6a and 6b r e p r e s e n t the e x a c t and approximate  solutions  of  the  the  F i g u r e 6c i s the p o i n t w i s e e r r o r in  this  near  figure  the  of  present  the  boundaries,  boundaries. type  that  This  end  conditions  experiments  conditions  vorticity  deserves  we  recomended i n  for have [4],  inthe  of t h e v o r t i c i t y .  particularly  fact  vorticity  errors the  more  the used  are  spline the  procedure. type  we have a l s o o b s e r v e d  n a t u r a l end c o n d i t i o n s gave v e r y poor r e s u l t s .  Ill  and  with  second  run.  We o b s e r v e  larger  eastern  testing  same  at  and  northern different In  the  of  end  that  the  h"  1  CFL  Vort.  Error  Str.Func. Error  20  1.27  -1.25  - 4 . 26  25  1.60  -1.42  -4.71  30  1. 90  - 1 . 55  - 5 . 01  35  2.22  - 1 . 70  - 5 . 51  40  2.37  - 1 . 90  -5.73  50  2.30  - 2 . 10  -6.20  T a b l e 3.  h"  Logarithm of  Re. 100  r.m.s.errors  CFL  Vort.  35  2. 22  -.63  - 3 . 50  40  2. 37  -.78  -3.77  II  50  2. 30  - . 90  - 4 . 20  II  1  T a b l e 4.  h"  1  Error Str.Func.  Logarithm of  CFL  Vort.  Error  r.m.s.  Error  Re.  errors  Str.Func.  Error  Re.  2. 22  -.36  - 2 . 50  40  2. 37  - . 48  - 2 . 81  II  50  2.30  -3.25  II  - . 65 Logarithm of  r.m.s.  112  Number  1000  35  T a b l e 5.  Number  errors  3000  Number  F i g . 5a E x a c t Stream F u n c t i o n  113  Solution  Fig.  5b  Approximated Stream  114  Function  5c P o i n t w i s e E r r o r f o r Stream  115  Function  Fl  9.  6a F y , „  +  116  117  118  APPENDIX  The  main  purpose  order  t o do  so  their  approximation  of  we n e e d some  The T r a c e  appendix  results  properties  are enunciated without  1.  this  TO CHAPTER  by  IV  is  to  prove  Lemma  2.3;  on t h e t r a c e o p e r a t o r s finite  elements.  These  in  y . and results  proof.  Theorem  2 Let  n  boundary £  1),  be  T.  a  bounded  We d e f i n e  Lipschitz  the  continuous  trace operators  subset  y ^ and y  of (j  R  an  with integer  f o r a f u n c t i o n u d e f i n e d i n Q by y u  = u|  Q  ,  r  (A. 1)  2  _ (A.2)  1=1  j  where  are  Theorem  Al.  number.  Then  can H  S  be  j. •  be  a  trace to  positive  defined  continuous the  y^., linear  mapping  and Magenes  consequence constants  c  of  of  [20 p. this  and c  above  j  < s  operators y 4.  H (Q)  as  0 s  operator rrS/n\  linear  a  the u n i t normal v e c t o r on  operators  Moreover,  Lions  of  domain  i-  continuous  As  SI  the  extended 2  Proof.  the components  Let  ^^ (D.  3  i  j  onto  :=  S-l/2  and  s  1/2,  j  mapping \t Q>  f|j n =  -  T.  a  integer,  H (Q)  VJ' ' ' '*p^  ..S-j-1/2 ^  such  119  that  onto i  s  a  . (F).  42] theorem  real  • we  have  that " there  exist  c . Hull _ _ £ lly .ull . _ _ £ c . llu II 1 s,2,Q j s-j-l/2,2,Q 2  The n o r m  in H  I f II  r-l/2  (D  r-l/2,2,Y  is  defined  _ _ =  inf  , • s,2,Q  (A.3)  n  by  \lv\l  _ _ . r , 2, fi  (A. 4 )  r v=f 0  In  we h a v e  Kerj^  2.  Boundary  Kerj  = H (Q)  the f a m i l y of  be a f i n i t e  finite  dimensional  H.  c  P,Cn;  dimensional  space  such  C  H, , n  1+1  the approximation p r o p e r t i e s  inf „ X*H  llu - *||  spaces  {Q^}-  that  . s, 2,  of  2: 02  ff^  £ 0  (Chap.II!  < Ch "lluII „ _ fi r , 2, fi  (A. 5)  0  h  a- = min( 1 + 1-s,  (A.5)  holds  for  Let  //Yn;  2  Recall  .  2  2  characterization  Families.  Consider e  the f o l l o w i n g  a r b i t r a r y r,  r-s)  s e v e n r,  120  s £ 0  (cf.  [1]).  Theorem A2.  Let  Q. be  functions  U e  fund  on T such  ions  a domain  with  property  For a proof  The  of  following  boundary \yj}J Q  spaces  (A.5)  > m -  this  there  be  given  following  i)  a  r  on  the  a  finite  ( £ h  w h e r e C is  H^(D  of  Aziz  [1].  properties  and  Schatz  of  [7].  o f o r d e r q^,  space  •  H^(Q),  the Let  and  let  then  the  such  that  0 £ 1 -  q . J  1/2,  0 £  J  J  lly .u - y .(/II  ;  <  J  J  a. be  h  V1/ J  W '  a real  J  number  1/2.  of  such  h and  y .  that  <7j_^  2/2  +  <  a  <  J  T h e n f o r u € /Y°Y£2;  lly .u - y l/ll . _ £ Ch *f ' J  (A. 6 )  2  independent  0 < u . £ 1 - q . J J  ,  J  J  a constant  Let  V j=0  Bramble  dimensional  numbers  j=o  h  to  J  i _ i  of  q .+u .  J  inf UeH  approximation  trace operators  1-1 q .+u .+A . C £ h lly ull  and  families  traces  - q . - u . J J  1-1  ii)  generate  the  > 0 .  due  e  u ., A . r e a l J J  A . £ J + 1/2 J  h  Then  i n e q u a l i t i e s hold:  For  inf  above.  theorem see Babuska-and  results H^CD  1/2  be a s y s t e m o f n o r m a l  =  as  that  1 + 1/2  Proof.  defined  3  2 r  121  oc-1/2 Hull a  '  _ _ . > 2  n  (A. 7 )  +  I  4 . P r o o f o f Lemma 2 . 3  To p r o v e  Lemma 2 . 3 we r e c a l l  s a t i s f i e s f o r q € H (Q)  f| H (Q)  S  (Vw, V )  v |  We f i r s t  r  - q  b  the R i t z  , V* 6 H  ,  Qh  P^g = w  (A.8.1)  (A.8.2)  .  h  e s t i m a t e the term llvYw - q)\\.  two a u x i l i a r y  projection  , 1 £ s £ 1+1  2  = CVg, V*;  X  that  Towards  t h i s e n d , we d e f i n e  problems.  L e t w, e H,(Q) 1  be the f i n i t e element a p p r o x i m a t i o n  h  -A V q  = f ,  2  H  u  q\  = q  T  h  of  ( A . 9 . 1)  •  (A.9.2)  Then  (Vw.,  where w  V) X  = CVg,  V*;  (Vw , Vx)  -  , *X e H , n  ,  (A.10)  e H (Q) i s s u c h t h a t w^lp = q . f o  L e t w^ be the f i n i t e element a p p r o x i m a t i o n  -V g' 2  g'l  r  = 0 ,  = q  (A.10.1)  - q  b h  of  h  (A.10.2]  .  Then (Vw  y  V) X  = -fVw , 4  Vx)  , Vx e H  122  Qh  ,  (A.11)  w h e r e w . € H (Q) i s s u c h t h a t w. |_ = q, , - q , . 4 4Y oh o By  the theory  Theor.  of finite  elements  f o re l l i p t i c  0  C depends  the t r a c e  on A . By v i r t u e n theorem  , w  Since  ( c f . [26],  8.5) we h a v e  liq - wJI. _ _ £ Ch "llqll . _ , cr = mind, 1 1,2,0. s, 2,Q  where  problems  3"  i C ] > 2 i n  using A. 6 with q  o f the Lax-Milgram  "b"Vi/2  q ^ i s thef i n i t e  s-1),  ,r  l 2  •  (A. 12)  theorem and  (A  -  13)  e l e m e n t a p p r o x i m a t i o n o f q ^ e H^Cfi),  = 0 and X^. = r  llvJI. , _ £ Chr~1/2\\q.\\ , _ i C_h .3 2,2, fi 2 ^b r+2/2,2, T 2  S_1  llqll  , _ . (A. 14) s, 2, fi  Now,  IIVCW  -q )\\2 = cvcw - q;, vcw - q;; = = CVCw - q;, ~V(V - w ) + V(w - q)) 1  £ IIVCw - q)\\ IIVCw - v ;il + HVfw - q)\\ IIVCw^- qjll . Hence  Wv  - q;il * II fv - V"2,2,fi = " 3 i,2,n w  From  then  (A.12) a n d (A.14),  U  +  i ~  +  nw  llw^ -  qH  *h.2.a  thef o l l o w i n g estimate  IIVCw - q;il £ Ch  S 2  llqll  s 2  fl  123  •  lf2>Q  holds  (A. 15)  It  remains  to  estimate  the  defining an auxiliary e l l i p t i c  term  llv  -  qll.  We proceed  by  problem with homogeneous boundary  2  2  conditions. Let \p e H (U) and tp e L (Q). Furthemore, -A V \I) n  = <p in f2 ,  2  U  (A. 16. 1)  0 1 = 0 .  (A.16.2)  By regularity theory  \\if>\l  22  n  s Cllf>ll .  (A. 18)  Then  Cw - q, <p) = -(w - q,V ip)  = CVCw - q), Vip) - <-|jj-p  2  For any x e  7 (^ ~ Q  HQh(Q)  (w - q, <p) = CVCw - q;, V C 0 -  -  <  -|  L n  '  -  q;>  p  (A.19) By virtue of the following items: i) Approximation properties of H ^  -  IVC0  ii)  s K hll^ll 2  2  n  = /C hll<pll 2  The trace theorem  l  l I/2,2 r V" 2 2,Q V" l  S  (  iii)  2  (A. 5)  l  S  )  Inequality A.6 and the trace theorem  124  where K_ are constants,  the inequality (A.19) becomes  (w - q, <p) £ K h\\V(w 6  - q)\\ llipll +K h \\q\\ S  7  s  2  Q  \\<p\\ .  Taking <p = w - q and using (A. 15) the Lemma is proven.  125  •  CHAPTER V  CONCLUSIONS  We have formulated and analysed an algorithm of Galerkin-Characteristic convection-dominated algorithm method,  has  formulation vorticity  diffusion  been  similar  combined  to  of  methods  the  the  one  to  problems.  with  a  given  in  function  integrate The  mixed  to  equations  equations)  proposed  finite  [11],  quasi-geostrophic  — stream  the type  for  element  produce  a  (potential  a mid—latitude  baroclinic ocean. The  integration  our algorithm  is  of  the  transport-diffusion  essentially  equations  a two stage process.  by  The f i r s t  stage corresponds to the integration of the advection operator along with  the  characteristic curves  Galerkin  method.  reinterpretation of -  in  terms  yields  of  For  this  the usual  particle  of  the  stage,  a computationally efficient  we  combination  show  with scheme,  functional of  the dependent variable at the departure The  scheme  the grid  maximum norm for  smooth functions at  the  foot  of  this  point  thus devised  unconditionally stable and convergent. the  grids,  is  values  of  of  a  points  conservative,  Our error analysis  in  stage proves that for sufficiently  the approximate solution is the  a  which consists  by  the particles.  that  rectangular  interpolating  of  splines  in  Galerkin-Characteristic method  methodology,  cubic  flow  characteristic  curves.  super convergent To  assess  the  performance of our algorithm for the hyperbolic stage we have  126  carried out two types of advective experiments.  The f i r s t one  is a f a i r l y hard problem which consists of advecting a cone in a rotating flow f i e l d .  Munz [28]  this  by  problem  obtained  high  has reported the results of resolution  finite  schemes of the type total variation diminishing  difference  (TVD)  such as  MUSCL, UNO, etc. , which are considered to be the best  finite  difference  visual  schemes  for  hyperbolic  problems.  A  comparison of our results with those portrayed in figures 6 to 16 of  [28] shows that:  i ) Our scheme is able to keep the shape of the cone better than any high resolution scheme. ii)  The 'numerical diffusion'  of our scheme is  lower than  that of the high resolution f i n i t e difference schemes, with the possible exception of the superbee scheme. iii)  Our scheme exhibits small wiggle activity at the base of  the  cone,  whereas  the  high  resolution  f i n i t e difference schemes are wiggle free because they possess the TVD property. iv) Our scheme is able to use much larger time steps than any high resolution f i n i t e ours  is  unconditionally  difference scheme because  stable  with  respect  to  the  2 L —norm. This property is  responsible  for keeping the  wiggles under s t r i c t control. Our  second  'slotted' hard  numerical  cylinder  and  the  consists  in a rotating flow f i e l d .  problem because  surface  experiment  of  the discontinuities  'slotted'  region,  127  of  advecting  This on  is  the  respectively.  A  a  a very lateral visual  comparison of our results with those reported in SHASTA and figures  6  modified SHASTA schemes, to  16,  using  the  and  [28],  aforementioned  [37],  using  specifically  high  resolution  schemes, yields the same conclusion as in the cone experiment. The second  stage corresponds  the algorithm via  to the time progression  of  the diffusion mechanism starting with the  output of the previous stage. out in our analysis  The time progression  by the Crank-Nicholson  scheme.  is carried The error  2 analysis  with  respect  to  L -norm,  based  on  developed in [10] and [35], reveals for At = 0(h) of a term of order 0(h)  techniques the presence  in the evolutionary component of the  error.  This term is due to the particle approximation of the  first  stage  and  is  suspected  Galerkin-Characteristic  methods  to  be  which  inherited  by  those  approximate  the  inner  products by alternative techniques, such as the ones proposed in  [27] and [5].  In this sense, the Crank—Nicholson scheme  is  suboptimal when used in conjunction with our method; however, for  larger time steps  it  leads to more accurate results  than  the backward Euler scheme , which appears to be optimal for At = 0(h).  Our error analysis  ones used in [13],  technique is more general than the  [31] and [33], which cannot be used for the  Crank—Nicholson scheme. The  constants  C  Q1  and  C  in  Q2  Chap.IV  (2.11),  which  multiply the terms of the evolutionary error, are the product of the exponential constant of the Gronwall's inequality with the  norm  of  characteristic  the  derivatives  curves.  For  of  the  convection  128  variable  along  the  dominated  flows  the  speed of  variation  along  the  characteristics  is  less  rapid  than in the t direction, so the algorithm permits larger time steps without loss of accuracy.  On the other hand,  since the  algorithm in the second stage is also unconditionally stable, then the exponential constant of the Gronwall's  inequality  is  less than one; so that as time progresses the influence of the evolutionary decreases,  error and  approximate  on  the  eventually  solution  is  global for  T  accuracy  of  the  — > oo the' error  the approximation  error.  method of  The  the  latter  2 type of error is 0(h ) with free-slip boundary conditions, and 0(h  ) with no—slip  boundary conditions.  one  order  than  higher  literature. with respect  Finally,  2  previous  we prove  to L -norm for  These estimates  estimates  that  the  given  approximation  the stream functions  is  2 which is the  optimal, while the L —norm approximation  velocities  is  0(h).  Numerical  in  experiments  the error  2  0(h ),  error  with  are  for  no-slip  boundary conditions shows the validity of our error estimates. As a f i n a l remark, we should mention that our analysis can be  extended  to  approximate  the  solution  equations by the proposed algorithm.  129  of  Navier—Stokes  REFERENCES  1. Adams,  R.A.:  Sobolev  Spaces.  Academic  Press,  New  York  (1975) 2.  Babuska,  I.  and  Aziz,  A.K.:  "Survey  Lectures  on  the  Mathematical Foundations of the Finite Element Method," Edi.,  The Mathematical  Foundations  Aziz,  A.K.,  Finite  Element Method with Applications  Differential  Equations.  of  to  Partial  Academeic Press, N.Y.  (1972)  in  the  pp 5-359 3.  Bardos,  C. , Bercovier, M. ,  and Pironneau,  0. :  The vortex  method with f i n i t e elements. Math. Comp. 36, 119-136 (1981) 4.  Behforooz, cubic  G. H.  and Papamichael,  splineinterpolation. J.  N. :  Inst.  End  Maths.  conditions  for  Applies.,  23,  355-366. (1979) 5.  Benque, A  J. 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(2)  For a proof of (2) see [18]. The  discrete  version  of  the  above  inequality  goes  as  follows.  Let  ( )' ^ ^' ^ r? a  C  n  n  b  e  ^  r e e  sequences  numbers such that (c ) is monotonically n a  +b £ n n  n-1  n  n *  of positive  increasing  1, A > 0 ,  real  and  (3)  then a  n  + b £ c exp(Xn) , n ^ 0 . n n  For a proof ao (4) see reference [15] Chap.V.  135  (4)  2.  Schwarz Inequality 2  2  Let n c R and f.geL  (Q), then  f fgdfi s f f d£> f g d n . 2  J r>  (5)  2  Jn  Jn  3. Young Inequality  Let e > 0, and a, b e R, then  ab £ -:—a 4c  +  eb .  (6)  4. Friedrichs Inequality  If Q is connected and bounded at least then for each integer  m £ 0, there exists  in one direct ion, a constant K = K(m,  Q) > 0, such that  Ivll  _ s K\v\ _ , Vv e fljjfn; . m  (7)  The proof of (7) can be found in [1]. A discrete version of this inequality is given in [38].  5. Lax-Milgram Theorem  Let a(u,v) be a continuous and elliptic V, i.e., \a(u,v)\ £ tfllull llvll  , Vu,  v e V  \a(u,u)\ £ allull ^ , Vu e V. 2  136  form on the space  Then  ,  the  problem  a(u,v)  has  a uniqe  Proof.  See  = <l,v>  solution  , Vv  in  e V and  V.  [2].  137  1 e  V  

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