Table Of ContentRobust
Computational
Techniques for
Boundary Layers
APPLIED MATHEMATICS
Editor: R.J. Knops
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1 Introduction to the Thermodynamics of Solids J.L. Ericksen (1991)
2 Order Stars A. Iserles and S. P. N0rsett (1991)
3 Material Inhomogeneities in Elasticity G. Maugin (1993)
4 Bivectors and Waves in Mechanics and Optics
Ph. Boulanger and M. Hayes (1993)
5 Mathematical Modelling of Inelastic Deformation
J.F. Besseling and E van der Geissen (1993)
6 Vortex Structures in a Stratified Fluid: Order from Chaos
Sergey I. Voropayev and Yakov D. Afanasyev (1994)
7 Numerical Hamiltonian Problems
J.M. Sanz-Sema and M.P Calvo (1994)
8 Variational Theories for Liquid Crystals E.G. Virga (1994)
9 Asymptotic Treatment of Differential Equations A. Georgescu (1995)
10 Plasma Physics Theory A. Sitenko and V. Malnev (1995)
11 Wavelets and Multiscale Signal Processing
A. Cohen and R.D. Ryan (1995)
12 Numerical Solution of Convection-Diffusion Problems
K.W. Morton (1996)
13 Weak and Measure-valued Solutions to Evolutionary PDEs
J. Malek, J. Necas, M. Rokyta and M. Ruzicka (1996)
14 Nonlinear Ill-Posed Problems
A.N. Tikhonov; A.S. Leonov andA.G. Yagola (1998)
15 Mathematical Models in Boundary Layer Theory
O.A. Oleinik and V.M. Samokhin (1999)
16 Robust Computational Techniques for Boundary Layers
P.A. Farrell, A.F. Hegarty, J.J.H. Miller,
E. OyRiordan and G. /. Shishkin (2000)
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Applied Mathematics 16
Robust
Computational
Techniques for
Boundary Layers
P. A. Farrell
Kent State University
A. F. Hegarty
Trinity College, Dublin
J. J. H. Miller
University of Limerick
E. O’Riordan
Dublin City University
G. I. Shishkin
Russian Academy of Sciences
Boca Raton London New York
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Library of Congress Cataloging-in-Publication Data
Robustcomputationaltechniquesforboundarylayers / P.A.Farrell...[etal.].
p.cm.(Appliedmathematicsand mathematicalcomputation ; 13)
Includesbibliographicalreferencesandindex.
ISBN 1-58488-192-5
I.Boundarylayer-Mathematics.2.Numericalcalculations. I.Farrell,P.A. II.Series.
QA913 .R62 2000
532'.05l---<ic2l 99-086205
CIP
LibraryofCongressCardNumber99-086205
To Avril, Kathy, Mary, Lida and Pamela
Contents
1 Introduction to numerical methods for problems
with boundary layers 1
1.1 The location and width of a boundary layer 1
1.2 Norms for boundary layer functions 3
1.3 Numerical methods 8
1.4 Robust layer-resolving methods 9
1.5 Some notation 11
2 Numerical methods on uniform meshes 13
2.1 Convection-diffusion problems in one dimension 13
2.2 Centred finite difference method 16
2.3 Monotone matrices and discrete comparison principles 19
2.4 Upwind finite difference methods 21
2.5 Fitted operator methods 26
2.6 Neumann boundary conditions 31
2.7 Error estimates in alternative norms 34
3 Layer resolving methods for convection diffusion
problems in one dimension 37
3.1 Bakhvalov fitted meshes 37
3.2 Piecewise-uniform fitted meshes 39
3.3 Theoretical results 44
3.4 Global accuracy on piecewise-uniform meshes 55
3.5 Approximation of derivatives 58
3.6 Alternative transition parameters 67
4 The limitations of non-monotone numerical
methods 73
4.1 Non-physical behaviour of numerical solutions 73
4.2 A non-monotone method 74
4.3 Accuracy and order of convergence 79
4.4 Tuning non-monotone methods 81
4.5 Neumann boundary conditions 87
4.6 Approximation of scaled derivatives 89
4.7 Further considerations 90
viii
5 Convection-diffusion problems in a moving medium 93
5.1 Motivation 93
5.2 Convection-diffusion problems 95
5.3 Location of regular and corner boundary layers 97
5.4 Asymptotic nature of boundary layers 100
5.5 Monotone parameter-uniform methods 104
5.6 Computed errors and computed orders of convergence 106
5.7 Numerical results 108
5.8 Neumann boundary conditions 109
5.9 Corner boundary layers 113
5.10 Computational work 118
6 Convection-diffusion problems with frictionless walls 121
6.1 The origin of parabolic boundary layers 121
6.2 Asymptotic nature 124
6.3 Inadequacy of uniform meshes 128
6.4 Fitted meshes for parabolic boundary layers 133
6.5 Simple parameter-uniform analytic approximations 140
7 Convection-diffusion problems with no slip
boundary conditions 147
7.1 No-slip boundary conditions 147
7.2 Width of degenerate parabolic boundary layers 150
7.3 Monotone fitted mesh method 151
7.4 Numerical results 152
7.5 Slip versus no-slip 153
8 Experimental estimation of errors 157
8.1 Theoretical error estimates 157
8.2 Quick algorithms 162
8.3 General algorithm 166
8.4 Validation 169
8.5 Practical uses of e-uniform error parameters 170
8.6 Global error parameters 171
9 Non—monotone methods in two dimensions 175
9.1 Non-monotone methods 175
9.2 Tuned non-monotone method 175
9.3 Difficulties in tuning non-monotone methods 182
9.4 Weaknesses of non-monotone e-uniform methods 189
10 Linear and nonlinear reaction—diffusion problems 191
10.1 Linear reaction diffusion problems 191
10.2 Semilinear reaction-diffusion problems 194
ix
10.3 Nonlinear solvers 195
10.4 Numerical methods on uniform meshes 197
10.5 Numerical methods on piecewise-uniform meshes 201
10.6 An alternative stopping criterion 206
11 Prandtl flow past a flat plate - Blasius’ method 209
11.1 Prandtl boundary layer equations 209
11.2 Blasius’ solution 212
11.3 Singularly perturbed nature of Blasius’ problem 213
11.4 Robust layer-resolving method for Blasius’ problem 214
11.5 Numerical solution of Blasius’ problem 216
11.6 Computed error estimates for Blasius’ problem 217
11.7 Computed global error estimates for Blasius’ solution 220
12 Prandtl flow past a flat plate - direct method 225
12.1 Prandtl problem in a finite domain 225
12.2 Nonlinear finite difference method 226
12.3 Solution of the nonlinear finite difference method 228
12.4 Error analysis based on the finest mesh solution 231
12.5 Error analysis based on the Blasius solution 234
12.6 A benchmark solution for laminar flow 247
References 249
Index 253
