Advances in computer technology have conveniently coincided withtrends in numerical analysis toward increased complexity ofcomputational algorithms based on finite difference methods. It isno longer feasible to perform stability investigation of thesemethods manually--and no longer necessary. As this book shows,modern computer algebra tools can be combined with methods fromnumerical analysis to generate programs that will do the jobautomatically.
Comprehensive, timely, and accessible--this is the definitivereference on the application of computerized symbolic manipulationsfor analyzing the stability of a wide range of difference schemes.In particular, it deals with those schemes that are used to solvecomplex physical problems in areas such as gas dynamics, heat andmass transfer, catastrophe theory, elasticity, shallow watertheory, and more.
Introducing many new applications, methods, and concepts,Computer-Aided Analysis of Difference Schemes for PartialDifferential Equations
* Shows how computational algebra expedites the task of stabilityanalysis--whatever the approach to stability investigation
* Covers ten different approaches for each stability method
* Deals with the specific characteristics of each method and itsapplication to problems commonly encountered by numerical modelers
* Describes all basic mathematical formulas that are necessary toimplement each algorithm
* Provides each formula in several global algebraic symboliclanguages, such as MAPLE, MATHEMATICA, and REDUCE
* Includes numerous illustrations and thought-provoking examplesthroughout the text
For mathematicians, physicists, and engineers, as well as forpostgraduate students, and for anyone involved with numericsolutions for real-world physical problems, this book provides avaluable resource, a helpful guide, and a head start ondevelopments for the twenty-first century.
Table of contents
The Necessary Basics from the Stability Theory of DifferenceSchemes and Polynomials.
Symbolic-Numerical Method for the Stability Investigation ofDifference Schemes on a Computer.
Application of Optimization Methods to the Stability Analysis ofDifference Schemes.
Stability Analysis of Difference Schemes by Catastrophe TheoryMethods.
Construction of Multiply Connected Stability Regions of DifferenceSchemes by Computer Algebra and Pattern Recognition.
Maximally Stable Difference Schemes.
Stability Analysis of Nonlinear Difference Schemes.
Symbolic Computation of Differential Approximations.