Robust Computational Techniques for Boundary Layers

by Farrell; Paul

• ISBN13: 9781584881926
• ISBN10: 1584881925
• Format: Hardcover
• Copyright: 2000-03-30
• Publisher: Chapman & Hall/

Summary

Current standard numerical methods are of little use in solving mathematical problems involving boundary layers. This volume presents an essentially new numerical technique that gives precise results in the boundary layer regions for the problems discussed. It shows that this technique can be adapted in a natural way to a real flow problem, and can also be used to construct benchmark solutions for comparison with solutions found using other numerical techniques.Written at the advanced undergraduate and research level, Robust Computational Techniques for Boundary Layers provides readers with an understanding of the underlying principles and the essential components needed for the construction of numerical methods for boundary layer problems.Marketing Class Code:

Author Biography

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

Table of Contents

Introduction to numerical methods for problems with boundary layers	1	(12)
The location and width of a boundary layer	1	(2)
Norms for boundary layer functions	3	(5)
Numerical methods	8	(1)
Robust layer--resolving methods	9	(2)
Some notation	11	(2)
Numerical methods on uniform meshes	13	(24)
Convection--diffusion problems in one dimension	13	(3)
Centred finite difference method	16	(3)
Monotone matrices and discrete comparison principles	19	(2)
Upwind finite difference methods	21	(5)
Fitted operator methods	26	(5)
Neumann boundary conditions	31	(3)
Error estimates in alternative norms	34	(3)
Layer resolving methods for convection diffusion problems in one dimension	37	(36)
Bakhvalov fitted meshes	37	(2)
Piecewise-uniform fitted meshes	39	(5)
Theoretical results	44	(11)
Global accuracy on piecewise-uniform meshes	55	(3)
Approximation of derivatives	58	(9)
Alternative transition parameters	67	(6)
The limitations of non-monotone numerical methods	73	(20)
Non-physical behaviour of numerical solutions	73	(1)
A non-monotone method	74	(5)
Accuracy and order of convergence	79	(2)
Tuning non-monotone methods	81	(6)
Neumann boundary conditions	87	(2)
Approximation of scaled derivatives	89	(1)
Further considerations	90	(3)
Convection-diffusion problems in a moving medium	93	(28)
Motivation	93	(2)
Convection-diffusion problems	95	(2)
Location of regular and corner boundary layers	97	(3)
Asymptotic nature of boundary layers	100	(4)
Monotone parameter-uniform methods	104	(2)
Computed errors and computed orders of convergence	106	(2)
Numerical results	108	(1)
Neumann boundary conditions	109	(4)
Corner boundary layers	113	(5)
Computational work	118	(3)
Convection-diffusion problems with frictionless walls	121	(26)
The origin of parabolic boundary layers	121	(3)
Asymptotic nature	124	(4)
Inadequacy of uniform meshes	128	(5)
Fitted meshes for parabolic boundary layers	133	(7)
Simple parameter-uniform analytic approximations	140	(7)
Convection-diffusion problems with no slip boundary conditions	147	(10)
No-slip boundary conditions	147	(3)
Width of degenerate parabolic boundary layers	150	(1)
Monotone fitted mesh method	151	(1)
Numerical results	152	(1)
Slip versus no-slip	153	(4)
Experimental estimation of errors	157	(18)
Theoretical error estimates	157	(5)
Quick algorithms	162	(4)
General algorithm	166	(3)
Validation	169	(1)
Practical uses of ε-uniform error parameters	170	(1)
Global error parameters	171	(4)
Non-monotone methods in two dimensions	175	(16)
Non-monotone methods	175	(1)
Tuned non-monotone method	175	(7)
Difficulties in tuning non-monotone methods	182	(7)
Weaknesses of non-monotone ε-uniform methods	189	(2)
Linear and nonlinear reaction-diffusion problems	191	(18)
Linear reaction diffusion problems	191	(3)
Semilinear reaction-diffusion problems	194	(1)
Nonlinear solvers	195	(2)
Numerical methods on uniform meshes	197	(4)
Numerical methods on piecewise-uniform meshes	201	(5)
An alternative stopping criterion	206	(3)
Prandl flow past a flat plate - Blasius' method	209	(16)
Prandlt boundary layer equations	209	(3)
Blasius' solution	212	(1)
Singularly perturbed nature of Blasius' problem	213	(1)
Robust layer-resolving method for Blasius' problem	214	(2)
Numerical solution of Blasius' problem	216	(1)
Computed error estimates for Blasius' problem	217	(3)
Computed global error estimates for Blasius' solution	220	(5)
Prandtl flow past a flat plate - direct method	225	(24)
Prandlt problem in a finite domain	225	(1)
Nonlinear finite difference method	226	(2)
Solution of the nonlinear finite difference method	228	(3)
Error analysis based on the finest mesh solution	231	(3)
Error analysis based on the Blasius solution	234	(13)
A benchmark solution for laminar flow	247	(2)
References	249	(4)
Index	253