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E-raamat: Time-Dependent Problems and Difference Methods, Second Edition 2nd Edition [Wiley Online]

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Time-Dependent Problems and Difference Methods, Second Edition 2nd EditionSuurem pilt
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Raamatu kodulehekülg: https://onlinelibrary.wiley.com/doi/book/10.1002/9781118548448
Gustofsson (emeritus, information technology, Uppsala U., Sweden), Kreiss (emeritus, mathematics, U. of California-Los Angeles), and computer scientist Oliger (1941-2006) discuss the derivation and analysis of numerical methods for computing approximate solutions to partial differential equations for time-dependent problems arising in the sciences and engineering. They write for graduate students interested in applied mathematics and scientific computation, as well as for physical scientists and engineers carrying out numerical experiments to investigate physical behavior and test designs. Even with added material this second edition is smaller than the first because they have simplified certain parts. Annotation ©2013 Book News, Inc., Portland, OR (booknews.com)

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.

Preface ix
Preface to the First Edition xi
Part I Problems With Periodic Solutions 1(246)
1 Model Equations
3(44)
1.1 Periodic Gridfunctions and Difference Operators,
3(7)
1.2 First-Order Wave Equation, Convergence, and Stability,
10(10)
1.3 Leap-Frog Scheme,
20(4)
1.4 Implicit Methods,
24(3)
1.5 Truncation Error,
27(3)
1.6 Heat Equation,
30(6)
1.7 Convection-Diffusion Equation,
36(3)
1.8 Higher Order Equations,
39(2)
1.9 Second-Order Wave Equation,
41(2)
1.10 Generalization to Several Space Dimensions,
43(4)
2 Higher Order Accuracy
47(18)
2.1 Efficiency of Higher Order Accurate Difference Approximations,
47(10)
2.2 Time Discretization,
57(8)
3 Well-Posed Problems
65(44)
3.1 Introduction,
65(5)
3.2 Scalar Differential Equations with Constant Coefficients in One Space Dimension,
70(2)
3.3 First-Order Systems with Constant Coefficients in One Space Dimension,
72(5)
3.4 Parabolic Systems with Constant Coefficients in One Space Dimension,
77(3)
3.5 General Systems with Constant Coefficients,
80(1)
3.6 General Systems with Variable Coefficients,
81(2)
3.7 Semibounded Operators with Variable Coefficients,
83(7)
3.8 Stability and Well-Posedness,
90(3)
3.9 The Solution Operator and Duhamel's Principle,
93(4)
3.10 Generalized Solutions,
97(2)
3.11 Well-Posedness of Nonlinear Problems,
99(3)
3.12 The Principle of A Priori Estimates,
102(5)
3.13 The Principle of Linearization,
107(2)
4 Stability and Convergence for Difference Methods
109(44)
4.1 The Method of Lines,
109(10)
4.2 General Fully Discrete Methods,
119(28)
4.3 Splitting Methods,
147(6)
5 Hyperbolic Equations and Numerical Methods
153(24)
5.1 Systems with Constant Coefficients in One Space Dimension,
153(3)
5.2 Systems with Variable Coefficients in One Space Dimension,
156(2)
5.3 Systems with Constant Coefficients in Several Space Dimensions,
158(2)
5.4 Systems with Variable Coefficients in Several Space Dimensions,
160(2)
5.5 Approximations with Constant Coefficients,
162(3)
5.6 Approximations with Variable Coefficients,
165(2)
5.7 The Method of Lines,
167(5)
5.8 Staggered Grids,
172(5)
6 Parabolic Equations and Numerical Methods
177(12)
6.1 General Parabolic Systems,
177(4)
6.2 Stability for Difference Methods,
181(8)
7 Problems with Discontinuous Solutions
189(58)
7.1 Difference Methods for Linear Hyperbolic Problems,
189(4)
7.2 Method of Characteristics,
193(6)
7.3 Method of Characteristics in Several Space Dimensions,
199(1)
7.4 Method of Characteristics on a Regular Grid,
200(8)
7.5 Regularization Using Viscosity,
208(2)
7.6 The Inviscid Burgers' Equation,
210(4)
7.7 The Viscous Burgers' Equation and Traveling Waves,
214(7)
7.8 Numerical Methods for Scalar Equations Based on Regularization,
221(6)
7.9 Regularization for Systems of Equations,
227(8)
7.10 High Resolution Methods,
235(12)
Part II Initial-boundary Value Problems 247(218)
8 The Energy Method for Initial-Boundary Value Problems
249(38)
8.1 Characteristics and Boundary Conditions for Hyperbolic Systems in One Space Dimension,
249(9)
8.2 Energy Estimates for Hyperbolic Systems in One Space Dimension,
258(8)
8.3 Energy Estimates for Parabolic Differential Equations in One Space Dimension,
266(5)
8.4 Stability and Well-Posedness for General Differential Equations,
271(3)
8.5 Semibounded Operators,
274(5)
8.6 Quarter-Space Problems in More than One Space Dimension,
279(8)
9 The Laplace Transform Method for First-Order Hyperbolic Systems
287(20)
9.1 A Necessary Condition for Well-Posedness,
287(4)
9.2 Generalized Eigenvalues,
291(1)
9.3 The Kreiss Condition,
292(3)
9.4 Stability in the Generalized Sense,
295(8)
9.5 Derivative Boundary Conditions for First-Order Hyperbolic Systems,
303(4)
10 Second-Order Wave Equations
307(32)
10.1 The Scalar Wave Equation,
307(17)
10.2 General Systems of Wave Equations,
324(3)
10.3 A Modified Wave Equation,
327(4)
10.4 The Elastic Wave Equations,
331(4)
10.5 Einstein's Equations and General Relativity,
335(4)
11 The Energy Method for Difference Approximations
339(38)
11.1 Hyperbolic Problems,
339(11)
11.2 Parabolic Problems,
350(7)
11.3 Stability, Consistency, and Order of Accuracy,
357(5)
11.4 SBP Difference Operators,
362(15)
12 The Laplace Transform Method for Difference Approximations
377(54)
12.1 Necessary Conditions for Stability,
377(10)
12.2 Sufficient Conditions for Stability,
387(18)
12.3 Stability in the Generalized Sense for Hyperbolic Systems,
405(11)
12.4 An Example that Does Not Satisfy the Kreiss Condition But is Stable in the Generalized Sense,
416(7)
12.5 The Convergence Rate,
423(8)
13 The Laplace Transform Method for Fully Discrete' Approximations
431(34)
13.1 General Theory for Approximations of Hyperbolic Systems,
431(20)
13.2 The Method of Lines and Stability in the Generalized Sense,
451(14)
Appendix A Fourier Series and Trigonometric Interpolation 465(12)
A.1 Some Results from the Theory of Fourier Series,
465(4)
A.2 Trigonometric Interpolation,
469(4)
A.3 Higher Dimensions,
473(4)
Appendix B Fourier and Laplace Transform 477(8)
B.1 Fourier Transform,
477(3)
B.2 Laplace Transform,
480(5)
Appendix C Some Results from Linear Algebra 485(4)
Appendix D SBP Operators 489(10)
References 499(8)
Index 507
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.
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