Ames / Rheinboldt / Jeffrey Numerical Methods for Partial Differential Equations

2. Auflage 2014
ISBN: 978-1-4832-6242-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 380 Seiten, Web PDF

Numerical Methods for Partial Differential Equations, Second Edition deals with the use of numerical methods to solve partial differential equations. In addition to numerical fluid mechanics, hopscotch and other explicit-implicit methods are also considered, along with Monte Carlo techniques, lines, fast Fourier transform, and fractional steps methods. Comprised of six chapters, this volume begins with an introduction to numerical calculation, paying particular attention to the classification of equations and physical problems, asymptotics, discrete methods, and dimensionless forms. Subsequent chapters focus on parabolic and hyperbolic equations, elliptic equations, and special topics ranging from singularities and shocks to Navier-Stokes equations and Monte Carlo methods. The final chapter discuss the general concepts of weighted residuals, with emphasis on orthogonal collocation and the Bubnov-Galerkin method. The latter procedure is used to introduce finite elements. This book should be a valuable resource for students and practitioners in the fields of computer science and applied mathematics.

Autoren/Hrsg.

Ames, William F.

Rheinboldt, Werner

Jeffrey, Alan

Inhaltsverzeichnis

1;Front Cover;1
2;Numerical Methods for Partial Differential Equations;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Preface to second edition;12
7;Preface to first edition;14
8;Chapter 1. Fundamentals;16
8.1;1-0 Introduction;16
8.2;1-1 Classification of physical problems;18
8.3;1-2 Classification of equations;20
8.4;1-3 Asymptotics;25
8.5;1-4 Discrete methods;29
8.6;1-5 Finite differences and computational molecules;30
8.7;1-6 Finite difference operators;34
8.8;1 -7 Errors;38
8.9;1-8 Stability and convergence;43
8.10;1-9 I rregular boundaries;45
8.11;1-10 Choice of discrete network;48
8.12;1-11 Dimensionless forms;49
8.13;REFERENCES;54
9;Chapter 2. Parabolic equations;56
9.1;2-0 Introduction;56
9.2;2-1 Simple explicit methods;57
9.3;2-2 Fourier stability method;62
9.4;2-3 Implicit methods;64
9.5;2-4 An unconditionally unstable difference equation;70
9.6;2-5 Matrix stability analysis;71
9.7;2-6 Extension of matrix stability analysis;74
9.8;2-7 Consistency, stability, and convergence;76
9.9;2-8 Pure initial value problems;77
9.10;2-9 Variable coefficients;79
9.11;2-10 Examples of equations with variable coefficients;83
9.12;2-11 General concepts of error reduction;85
9.13;2-12 Explicit methods for nonlinear problems;88
9.14;2-13 An application of the explicit method;92
9.15;2-14 Implicit methods for nonlinear problems;97
9.16;2-15 Concluding remarks;104
9.17;REFERENCES;105
10;Chapter 3. Elliptic equations;107
10.1;3-0 Introduction;107
10.2;3-1 Simple finite difference schemes;109
10.3;3-2 Iterative methods;113
10.4;3-3 Linear elliptic equations;115
10.5;3-4 Some point iterative methods;118
10.6;3-5 Convergence of point iterative methods;122
10.7;3-6 Rates of convergence;129
10.8;3-7 Accelerations—successive over-relaxation (SOR);134
10.9;3-8 Extensions of SOR;140
10.10;3-9 Qualitative examples of over-relaxation;145
10.11;3-10 Other point iterative methodsf;150
10.12;3-11 Block iterative methods;159
10.13;3-12 Alternating direction methods;163
10.14;3-13 Summary of ADI results;167
10.15;3-14 Some nonlinear examples;173
10.16;REFERENCES;176
11;Chapter 4. Hyperbolic equations;180
11.1;4-0 Introduction;180
11.2;4-1 The quasilinear system;185
11.3;4-2 Introductory examples;191
11.4;4-3 Method of characteristics;195
11.5;4-4 Constant states and simple waves;200
11.6;4-5 Typical application of characteristics;201
11.7;4-6 Explicit finite difference methods;208
11.8;4-7 Overstability;212
11.9;4-8 Implicit methods for second-order equations;214
11.10;4-9 Nonlinear examples;216
11.11;4-10 Simultaneous first-order equations—explicit methods;218
11.12;4-11 An implicit method for first-order equations;224
11.13;4-11 An implicit method for first-order equations;224
11.14;4-13 Gas dynamics in one-space variable;227
11.15;4-14 Eulerian difference equations;229
11.16;4-15 Lagrangian difference equations;234
11.17;4-15 Lagrangian difference equations;234
11.18;4-16 Hopscotch methods for conservation laws;236
11.19;4-17 Explicit-implicit schemes for conservation laws;239
11.20;REFERENCES;242
12;Chapter 5. Special topics;245
12.1;5-0 Introduction;245
12.2;5-1 Singularities;245
12.3;5-2 Shocks;253
12.4;5-3 Eigenvalue problems;259
12.5;5-4 Parabolic equations in several space variables;266
12.6;5-5 Additional comments on elliptic equations;270
12.7;5-6 Hyperbolic equations in higher dimensions;277
12.8;5-7 Mixed systems;285
12.9;5-8 Higher-order equations in elasticity and vibrations;289
12.10;5-9 Fluid mechanics: the Navier-Stokes equations;296
12.11;5-10 Introduction to Monte Carlo methods;314
12.12;5-11 Method of lines;317
12.13;5-12 Fast Fourier transform and applications;319
12.14;5-13 Method of fractional steps;322
12.15;REFERENCES;326
13;Chapter 6. Weighted residuals and finite elements;335
13.1;6-0 Introduction;335
13.2;6-1 Weighted residual methods (WRM);335
13.3;6-2 Orthogonal collocation;340
13.4;6-3 Bubnov-Galerkin (B-G) method;344
13.5;6-4 Remarks on completeness, convergence, and error bounds;348
13.6;6-5 Nagumo's lemma and application;354
13.7;6-6 Introduction to finite elements;357
13.8;REFERENCES;363
14;Author Index;366
15;Subject Index;372