Time–Dependent Problems and Difference Methods

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.

Written by authors who are at the forefront of the field, this book is divided into two parts: Part I discusses problems with periodic solutions, and Part II, which has been completely rewritten,  discusses initial boundary value problems for partial differential equations and numerical methods for them.  The problems with periodic solutions have been chosen since they allow the application of Fourier analysis without the complication that arises from the infinite domain for the corresponding Cauchy problem.  Furthermore, the analysis of periodic problems provides necessary conditions when constructing methods for initial boundary value problems.  Much of the material included in Part II appeared for the first time in the First Edition of this book.  The authors draw on their own interests and combined extensive experience in applied mathematics and computer science to bring about this practical and useful guide.  They also provide complete discussions of the pertinent theorems and back them up with examples and illustrations.  The book covers the analysis of numerical methods for computing approximate solutions to partial differential equations for time dependent problems.  The Second Edition features new coverage on hyperbolic first order systems on structured grids with a focus on the boundaries treated by special difference approximations.  This approach  allows the difference methods to be very competitive compared to methods on unstructured grids.  This new edition also contains new coverage of artificial boundary conditions and how to treat problems where the computational domain has been truncated, i.e. for external flow problems.  This is an important issue from an application point of view, but no satisfactory coherent presentation currently exists.  This original book includes a concrete discussion of initial boundary value problems for partial differential equations.  The authors have redone many of these results especially for this volume, including theorems, examples, and over one hundred illustrations.

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.