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9780849373794

Numerical Solutions for Partial Differential Equations: Problem Solving Using Mathematica

by ;
  • ISBN13:

    9780849373794

  • ISBN10:

    0849373794

  • Edition: Disk
  • Format: Hardcover
  • Copyright: 1996-07-12
  • Publisher: CRC Press

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Summary

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.

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.

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