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A symbolic approach to generation and analysis of finite difference schemes of partial differential equations. (English) Zbl 1250.65109

Langer, Ulrich (ed.) et al., Numerical and symbolic scientific computing. Progress and prospects. New York, NY: Springer (ISBN 978-3-7091-0793-5/pbk; 978-3-7091-0794-2/ebook). Texts & Monographs in Symbolic Computation, 123-156 (2012).
Summary: We discuss three symbolic approaches for the generation of a finite difference scheme of a general single partial differential equation (PDE). We concentrate on the case of a linear PDE with constant coefficients and prove, that these three approaches are equivalent. We systematically use another symbolic technique, namely the cylindrical algebraic decomposition, in order to derive conditions for the von Neumann stability of a given difference scheme. We demonstrate algorithmic symbolic approaches for the computation of both continuous resp. discrete dispersion relations of a linear PDE with constant coefficients resp. a finite difference scheme. We present an implementation of tools for the generation of schemes in the computer algebra system Singular. Numerous examples are computed with our implementation and presented in details. Some of the methods we propose can be generalized to nonlinear PDEs as well as to the case of variable coefficients and to the case of systems of equations.
For the entire collection see [Zbl 1234.65014].

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L05 Wave equation
65Y15 Packaged methods for numerical algorithms
68W30 Symbolic computation and algebraic computation

Software:

FIDE; SINGULAR; REDLOG; QEPCAD
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