Computer-Aided Analysis of Difference Schemes for Partial Differential EquationsAdvances 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. |
Contents
The Necessary Basics from the Stability Theory of Difference | 1 |
46 | 21 |
References | 67 |
SymbolicNumerical Method for the Stability Investigation | 77 |
References | 156 |
Stability Analysis of Difference Schemes by Catastrophe Theory | 199 |
References | 237 |
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Common terms and phrases
a₁ accuracy advection equation algorithm analysis of difference analytic application axis catastrophe theory Cauchy problem cell characteristic polynomial coefficients Computational Physics computer algebra consider coordinates corresponding curve denote determined difference equations difference schemes difference schemes approximating differential approximation domain example expression finite difference schemes fluid formula FORTRAN Fourier fractional steps function Ganzha gradient grid h₁ h₂ hyperbolic inequality K₁ K₂ linear MacCormack scheme mathematical matrix maximally stable difference Mazurik multiply connected N₁ Nauka Navier-Stokes equations Neumann nodes nondimensional nondimensional complexes nth level Numerical Methods numerical solution obtained operator optimization parameters partial differential equations Peyret plane points present problem quantities random variable result Richtmyer Russian satisfied Section Shokin spatial stability analysis stability condition stability investigation stability region boundary stable difference schemes stencil symbolic-numerical method Taylor series two-dimensional u₁ vector Vorozhtsov zeros ди