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More About This Textbook
Overview
Partial differential equations (PDEs) play an important role in the natural sciences and technology, because they describe the way systems (natural and other) behave. The inherent suitability of PDEs to characterizing the nature, motion, and evolution of systems, has led to their wide-ranging use in numerical models that are developed in order to analyze systems that are not otherwise easily studied. Numerical Solutions for Partial Differential Equations contains all the details necessary for the reader to understand the principles and applications of advanced numerical methods for solving PDEs. In addition, it shows how the modern computer system algebra Mathematica® can be used for the analytic investigation of such numerical properties as stability, approximation, and dispersion.
Editorial Reviews
Booknews
Based on lecture courses in Germany and Russia, explains the general concepts behind partial differential equations, and encourages the manipulation of them and the generation of new ideas using the commercially available software package Mathematica. Assumes no previous exposure to numerical methods. The symbols and codes used throughout are those of a personal computer rather than of pencil and paper. The 3.5" disk for Microsoft Windows contains all the programs described. Annotation c. Book News, Inc., Portland, OR (booknews.com)Product Details
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Table of Contents
Introduction to Mathematica General Information about Mathematica Symbolic Computations with Mathematica Numerical Computations with Mathematica Finite Difference Methods for Hyperbolic PDEs Construction of Difference Schemes for the Advection Equation The Notion of Approximation Fourier Stability Analysis Elementary Second-Order Schemes Algorithm for Automatic Determination of Approximation Order of Scalar Difference Schemes Monotonicity Property of Difference Schemes TVD Schemes The Construction of Difference Schemes for Systems of PDEs Implicit Difference Schemes Von Neumann Stability Analysis in the Case of Systems of Difference Equations Difference Initial- and Boundary-Value Problems Construction of Difference Schemes for Multidimensional Hyperbolic Problems Determination of Planar Stability Regions Curvilinear Spatial Grids Answers to the Exercises Finite Difference Methods for Parabolic PDEs Basic Types of Boundary Conditions for Parabolic PDEs Simple Schemes for the One-Dimensional Heat Equation Difference Schemes for Advection-Diffusion Equation Runge-Kutta Methods Finite Volume Method The Adi Method Approximate Factorization Scheme Dispersion Answers to the Exercises Numerical Methods for Elliptic PDEs Boundary-Value Problems for Elliptic PDEs A Simple Elliptic Solver Pseudo-Unsteady Methods The Finite Element Method Numerical Grid Generation Local Approximation Study of Finite Volume Operators on Arbitrary Grids Local Approximation Study of Difference Schemes on Logically Rectangular Grids Answers to the Exercises Appendix Glossary of Programs Index Each Chapter also includes a list of references.