Many physical problems involve diffusive and convective (transport) processes. When diffusion dominates convection, standard numerical methods work satisfactorily. But when convection dominates diffusion, the standard methods become unstable, and special techniques are needed to compute accurate numerical approximations of the unknown solution. This convection-dominated regime is the focus of the book. After discussing at length the nature of solutions to convection-dominated convection-diffusion problems, the authors motivate and design numerical methods that are particularly suited to this class of problems.
At first they examine finite-difference methods for two-point boundary value problems, as their analysis requires little theoretical background. Upwinding, artificial diffusion, uniformly convergent methods, and Shishkin meshes are some of the topics presented. Throughout, the authors are concerned with the accuracy of solutions when the diffusion coefficient is close to zero. Later in the book they concentrate on finite element methods for problems posed in one and two dimensions.
This lucid yet thorough account of convection-dominated convection-diffusion problems and how to solve them numerically is meant for beginning graduate students, and it includes a large number of exercises. An up-to-date bibliography provides the reader with further reading.
Author(s): Martin Stynes, David Stynes
Series: Graduate Studies in Mathematics 196
Edition: 1
Publisher: American Mathematical Society
Year: 2018
Language: English
Pages: 164
Tags: Convection-Diffusion Equations
Title page 4
Preface 8
Chapter 1. Introduction and Preliminary Material 10
1.1. A simple example 10
1.2. A little motivation and history 16
1.3. Notation 16
1.4. Maximum principle and barrier functions 17
1.5. Asymptotic expansions 19
Chapter 2. Convection-Diffusion Problems in One Dimension 24
2.1. Asymptotic analysis—an extended example 24
2.2. Green’s functions 30
2.3. A priori bounds on the solution and its derivatives 33
2.4. Decompositions of the solution 47
Chapter 3. Finite Difference Methods in One Dimension 52
3.1. M-matrices, upwinding 54
3.2. Artificial diffusion 63
3.3. Uniformly convergent schemes 65
3.4. Shishkin meshes 68
Chapter 4. Convection-Diffusion Problems in Two Dimensions 78
4.1. General description 78
4.2. A priori estimates 86
4.3. General comments on numerical methods 93
Chapter 5. Finite Difference Methods in Two Dimensions 96
5.1. Extending one-dimensional approaches 96
5.2. Shishkin meshes 98
5.3. Characteristic boundary layers 100
5.4. Other remarks 102
Chapter 6. Finite Element Methods 104
6.1. The loss of stability in the (Bubnov–)Galerkin FEM 104
6.2. Relationship to classical FEM analysis 107
6.3. ?*-splines 109
6.4. The streamline-diffusion finite element method (SUPG) 112
6.5. Stability of the Galerkin FEM for higher-degree polynomials 120
6.6. Shishkin meshes 126
6.7. Discontinuous Galerkin finite element method 135
6.8. Continuous interior penalty (CIP) method 140
6.9. Adaptive methods 148
Chapter 7. Concluding Remarks 152
Bibliography 154
Index 164
