Bertil Gustafsson, Heinz-Otto Kreiss, Joseph Oliger

Time-Dependent Problems 2e

• Gebundenes Buch

Produktbeschreibung

Written by authors at the forefront of their field, this Second Edition discusses problems with periodic solutions, and presents new information on initial boundary value problems and numerical methods for partial differential equations. Complete with reworked theorems, examples, and illustrations, it covers hyperbolic first order systems on structured grids; differential equations and numerical methods with a parallel development; hyperbolic equations; error bounds and estimates; artificial boundary conditions; and more. This book is an ideal reference for anyone involved in physical science, engineering, numerical analysis, and mathematical modeling.

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Praise for the First Edition

". . . fills a considerable gap in the numerical analysis literature by providing a self-contained treatment . . . this is an important work written in a clear style . . . warmly recommended to any graduate student or researcher in the field of the numerical solution of partial differential equations."
--SIAM Review

Time-Dependent Problems and Difference Methods, Second Edition continues to provide guidance for the analysis of difference methods for computing approximate solutions to partial differential equations for time-dependent problems. The book treats differential equations and difference methods with a parallel development, thus achieving a more useful analysis of numerical methods.

The Second Edition presents hyperbolic equations in great detail as well as new coverage on second-order systems of wave equations including acoustic waves, elastic waves, and Einstein equations. Compared to first-order hyperbolic systems, initial-boundary value problems for such systems contain new properties that must be taken into account when analyzing stability. Featuring the latest material in partial differential equations with new theorems, examples, and illustrations,Time-Dependent Problems and Difference Methods, Second Edition also includes:
High order methods on staggered grids
Extended treatment of Summation By Parts operators and their application to second-order derivatives
Simplified presentation of certain parts and proofs

Time-Dependent Problems and Difference Methods, Second Edition is an ideal reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs and to predict and investigate physical phenomena. The book is also excellent for graduate-level courses in applied mathematics and scientific computations.

Produktdetails

• Wiley Series in Pure and Applied Mathematics
• Verlag: Wiley & Sons
• 2. Aufl.
• Seitenzahl: 528
• Erscheinungstermin: 5. August 2013
• Englisch
• Abmessung: 240mm x 161mm x 33mm
• Gewicht: 848g
• ISBN-13: 9780470900567
• ISBN-10: 0470900563
• Artikelnr.: 37327543

Autorenporträt

BERTIL GUSTAFSSON, PhD, is Professor Emeritus in the Department of Information Technology at Uppsala University and is well known for his work in initial-boundary value problems. HEINZ-OTTO KREISS, PhD, is Professor Emeritus in the Department of Mathematics at University of California, Los Angeles and is a renowned mathematician in the field of applied mathematics. JOSEPH OLIGER, PhD, was Professor in the Department of Computer Science at Stanford University and was well known for his early research in numerical methods for partial differential equations.

Inhaltsangabe

Preface ix Preface to the First Edition xi PART I PROBLEMS WITH PERIODIC
SOLUTIONS 1 1. Model Equations 3 1.1. Periodic Gridfunctions and Difference
Operators 3 1.2. First-Order Wave Equation Convergence and Stability 10
1.3. Leap-Frog Scheme 20 1.4. Implicit Methods 24 1.5. Truncation Error 27
1.6. Heat Equation 30 1.7. Convection-Diffusion Equation 36 1.8. Higher
Order Equations 39 1.9. Second-Order Wave Equation 41 1.10. Generalization
to Several Space Dimensions 43 2. Higher Order Accuracy 47 2.1. Efficiency
of Higher Order Accurate Difference Approximations 47 2.2. Time
Discretization 57 3. Well-Posed Problems 65 3.1. Introduction 65 3.2.
Scalar Differential Equations with Constant Coefficients in One Space
Dimension 70 3.3. First-Order Systems with Constant Coefficients in One
Space Dimension 72 3.4. Parabolic Systems with Constant Coefficients in One
Space Dimension 77 3.5. General Systems with Constant Coefficients 80 3.6.
General Systems with Variable Coefficients 81 3.7. Semibounded Operators
with Variable Coefficients 83 3.8. Stability and Well-Posedness 90 3.9. The
Solution Operator and Duhamel's Principle 93 3.10. Generalized Solutions 97
3.11. Well-Posedness of Nonlinear Problems 99 3.12. The Principle of A
Priori Estimates 102 3.13. The Principle of Linearization 107 4. Stability
and Convergence for Difference Methods 109 4.1. The Method of Lines 109
4.2. General Fully Discrete Methods 119 4.3. Splitting Methods 147 5.
Hyperbolic Equations and Numerical Methods 153 5.1. Systems with Constant
Coefficients in One Space Dimension 153 5.2. Systems with Variable
Coefficients in One Space Dimension 156 5.3. Systems with Constant
Coefficients in Several Space Dimensions 158 5.4. Systems with Variable
Coefficients in Several Space Dimensions 160 5.5. Approximations with
Constant Coefficients 162 5.6. Approximations with Variable Coefficients
165 5.7. The Method of Lines 167 5.8. Staggered Grids 172 6. Parabolic
Equations and Numerical Methods 177 6.1. General Parabolic Systems 177 6.2.
Stability for Difference Methods 181 7. Problems with Discontinuous
Solutions 189 7.1. Difference Methods for Linear Hyperbolic Problems 189
7.2. Method of Characteristics 193 7.3. Method of Characteristics in
Several Space Dimensions 199 7.4. Method of Characteristics on a Regular
Grid 200 7.5. Regularization Using Viscosity 208 7.6. The Inviscid Burgers'
Equation 210 7.7. The Viscous Burgers' Equation and Traveling Waves 214
7.8. Numerical Methods for Scalar Equations Based on Regularization 221
7.9. Regularization for Systems of Equations 227 7.10. High Resolution
Methods 235 PART II INITIAL-BOUNDARY VALUE PROBLEMS 247 8. The Energy
Method for Initial-Boundary Value Problems 249 8.1. Characteristics and
Boundary Conditions for Hyperbolic Systems in One Space Dimension 249 8.2.
Energy Estimates for Hyperbolic Systems in One Space Dimension 258 8.3.
Energy Estimates for Parabolic Differential Equations in One Space
Dimension 266 8.4. Stability and Well-Posedness for General Differential
Equations 271 8.5. Semibounded Operators 274 8.6. Quarter-Space Problems in
More than One Space Dimension 279 9. The Laplace Transform Method for
First-Order Hyperbolic Systems 287 9.1. A Necessary Condition for
Well-Posedness 287 9.2. Generalized Eigenvalues 291 9.3. The Kreiss
Condition 292 9.4. Stability in the Generalized Sense 295 9.5. Derivative
Boundary Conditions for First-Order Hyperbolic Systems 303 10. Second-Order
Wave Equations 307 10.1. The Scalar Wave Equation 307 10.2. General Systems
of Wave Equations 324 10.3. A Modified Wave Equation 327 10.4. The Elastic
Wave Equations 331 10.5. Einstein's Equations and General Relativity 335
11. The Energy Method for Difference Approximations 339 11.1. Hyperbolic
Problems 339 11.2. Parabolic Problems 350 11.3. Stability Consistency and
Order of Accuracy 357 11.4. SBP Difference Operators 362 12. The Laplace
Transform Method for Difference Approximations 377 12.1. Necessary
Conditions for Stability 377 12.2. Sufficient Conditions for Stability 387
12.3. Stability in the Generalized Sense for Hyperbolic Systems 405 12.4.
An Example that Does Not Satisfy the Kreiss Condition But is Stable in the
Generalized Sense 416 12.5. The Convergence Rate 423 13. The Laplace
Transform Method for Fully Discrete Approximations 431 13.1. General Theory
for Approximations of Hyperbolic Systems 431 13.2. The Method of Lines and
Stability in the Generalized Sense 451 Appendix A Fourier Series and
Trigonometric Interpolation 465 A.1. Some Results from the Theory of
Fourier Series 465 A.2. Trigonometric Interpolation 469 A.3. Higher
Dimensions 473 Appendix B Fourier and Laplace Transform 477 B.1. Fourier
Transform 477 B.2. Laplace Transform 480 Appendix C Some Results from
Linear Algebra 485 Appendix D SBP Operators 489 References 499 Index 507