Abstract
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.
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Notes
- 1.
Used in the discretization of the advection equation
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Acknowledgements
The authors express their deep gratitude to Vladimir P. Gerdt (JINR, Russia) for his interest, discussions and suggestions during the work on this paper. We would also like to thank M. Kauers (RISC, Linz, Austria), W. Zulehner (J. Kepler University of Linz, Austria), M. Fröhner (BTU Cottbus, Germany) and A. Klar (TU Kaiserslautern, Germany) for discussions on various topics around stability in this paper. We have learned many examples from the scripts and papers of colleagues, mentioned above. A special thanks goes to H. Engl (Vienna, Austria) for his constructive critics, which helped to improve the presentation of the results. At last, but not at least, we thank to anonymous referees for their remarks and questions.
The first author is grateful to the SFB F013 “Numerical and Symbolic Scientific Computing” of the Austrian FWF for partial financial support in 2005-2007.
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Levandovskyy, V., Martin, B. (2012). A Symbolic Approach to Generation and Analysis of Finite Difference Schemes of Partial Differential Equations. In: Langer, U., Paule, P. (eds) Numerical and Symbolic Scientific Computing. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0794-2_7
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