Robust Computational Techniques for Boundary Layers APPLIED MATHEMATICS Editor: R.J. Knops This series presents texts and monographs at graduate and research level covering a wide variety of topics of current research interest in modem and traditional applied mathematics, in numerical analysis and computation. 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) (Full details concerning this series, and more information on titles in preparation are available from the publisher.) 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 CRC Press is an imprint of the Taylor & Francis Group, an informa business A CHAPMAN & HALL BOOK Firstpublished2000byCRCPress Published2018byCRCPress Taylor&FrancisGroup 6000BrokenSoundParkwayNW,Suite300 BocaRaton,FL33487-2742 ©2000byTaylor&FrancisGroup,LLC CRCPressisanimprintofTaylor&FrancisGroup,anlnformabusiness NoclaimtooriginalU.S.Governmentworks ISBN13: 978-1-58488-192-6(hbk) Thisbookcontainsinformationobtainedfromauthenticandhighlyregardedsources.Reasonableefforts have beenmade topublishreliabledata andinformation, butthe authorandpublishercannotassume responsibilityforthevalidityofallmaterialsortheconsequencesoftheiruse.Theauthorsandpublishers haveattemptedtotracethecopyrightholdersofallmaterialreproducedinthispublicationandapologize tocopyrightholdersifpermissiontopublishinthisformhasnotbeenobtained.Ifanycopyrightmaterial hasnotbeenacknowledgedpleasewriteandletusknowsowemayrectifyinanyfuturereprint. 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Visit the Taylor &Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 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