Screen LibraryNonstandard finite difference models of differential equations /
This book provides a clear summary of the work of the author on the construction of nonstandard finite difference schemes for the numerical integration of differential equations. The major thrust of the book is to show that discrete models of differential equations exist such that the elementary typ...
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| Main Authors: | |
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| Corporate Authors: | |
| Published: |
World Scientific Pub. Co.,
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| Publisher Address: | Singapore ; River Edge, N.J. : |
| Publication Dates: | 1994. |
| Literature type: | eBook |
| Language: | English |
| Subjects: | |
| Online Access: |
http://www.worldscientific.com/worldscibooks/10.1142/2081#t=toc |
| Summary: |
This book provides a clear summary of the work of the author on the construction of nonstandard finite difference schemes for the numerical integration of differential equations. The major thrust of the book is to show that discrete models of differential equations exist such that the elementary types of numerical instabilities do not occur. A consequence of this result is that in general bigger step-sizes can often be used in actual calculations and/or finite difference schemes can be constructed that are conditionally stable in many instances whereas in using standard techniques no such schemes exist. The theoretical basis of this work is centered on the concepts of "exact" and "best" finite difference schemes. In addition, a set of rules is given for the discrete modeling of derivatives and nonlinear expressions that occur in differential equations. These rules often lead to a unique nonstandard finite difference model for a given differential equation. |
| Carrier Form: | 1 online resource (xi,249pages) : illustrations |
| Bibliography: | Includes bibliographical references and index. |
| ISBN: | 9789814440882 |
| Index Number: | QA431 |
| CLC: | O175 |
| Contents: | 1 Introduction. 1.1. Exploiting parallelism. 1.2. Parallel computers. 1.3. Performance evaluations -- 2. Parallel algorithms for solving PDE. 2.1. Discretization by finite difference method. 2.2. Parallel relaxation algorithms. 2.3. Parallel ADI algorithm. 2.4. Parallel multigrid method. 2.5. Parallel conjugate gradient algorithm -- 3. Implementations. 3.1. Intel iPSC/860 hypercubes. 3.2. Inter-processor communications on iPSC/860. 3.3. Communication analysis for domain decomposition method. 3.4. Bandwidth improvement using forced message type. 3.5. KSR-1 parallel computers. 3.6. Automatic, semi-automatic and manual parallelization -- 4. Applications. 4.1. Numerical solution of Poisson's equation. 4.2. Numerical simulations of multi-phase flow. 4.3. Discretization and parallelization. 4.4. Numerical experiments -- 5. Parallel time stepping algorithms. 5.1. A new dimension for parallelization. 5.2. Waveform relaxation. 5.3. Pipeline iteration. 5.4. Window relaxation. 5.5. Parabolic multigrid method. 5.6. General form of parallel time stepping algorithms. 5.7. Complexity analysis -- 6. Future development. 6.1. More on performance evaluations. 6.2. Programming language. 6.3. Automatic parallelization tools. 6.4. Distributed computing network. 6.5. Metacenters and metacomputers. |