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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Used in the discretization of the advection equation
References
Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics. 10. Springer (2003)
Brown, C.W.: QEPCAD B: A program for computing with semi-algebraic sets using CADs. SIGSAM Bull. 37(4), 97–108 (2003). DOI 10.1145/968708.968710. http://www.usna.edu/Users/cs/qepcad/B/QEPCAD.html
Chyzak, F., Quadrat, A., Robertz, D.: Linear control systems over Ore algebras. effective algorithms for the computation of parametrizations. Applicable Algebra in Engineering, Communication and Computing 16(5), 938–1279 (2005). http://www.springerlink.com/content/y61643p573387258
Chyzak, F., Salvy, B.: Non-commutative elimination in Ore algebras proves multivariate identities. J. Symbolic Comput. 26(2), 187–227 (1998)
Cohn, R.M.: Difference algebra. R.E. Krieger (1979)
Decker, W., Greuel, G.M., Pfister, G., Schönemann, H.: Singular3-1-2 — A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2010). http://www.singular.uni-kl.de
Dolzmann, A., Sturm, T.: Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31(2), 2–9 (1997). http://redlog.dolzmann.de
Fröhner, M.: Numerische Methoden in der Hydrodynamik. Wiss. Schriftenr. Tech. Hochsch. Karl-Marx-Stadt 12 (1984)
Ganzha, V., Vorozhtsov, E.: Computer-Aided Analysis of Difference Schemes for Partial Differential Equations. Wiley Interscience (1996)
Ganzha, V.G., Vorozhtsov, E.V.: Parallel implementation of stability analysis of difference schemes with Mathematica. J. Math. Sci. 108, 1070–1088 (2002). DOI 10.1023/A:1013500723898. http://dx.doi.org/10.1023/A:1013500723898
Gerdt, V.: Involutive algorithms for computing Groebner bases. In: Pfister, G., Cojocaru, S., Ufnarovski V. (eds.) Computational Commutative and Non-Commutative Algebraic Geometry. IOS Press (2005)
Gerdt, V., Blinkov, Y.: Involution and difference schemes for the Navier-Stokes equations. Proceedings CASC 2009, Kobe, Japan (2009)
Gerdt, V., Blinkov, Y., Mozzhilkin, V.: Gröbner bases and generation of difference schemes for partial differential equations. SIGMA 2, 051 (2006). http://arxiv.org/abs/math/0605334
Gerdt, V., Robertz, D.: Consistency of finite difference approximations for linear PDE systems and its algorithmic verification. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC’10). ACM Press (2010). DOI 10.1145/1837934.1837950
Greuel, G.M., Levandovskyy, V., Motsak, A., Schönemann, H.: Plural. A Singular3.1 Subsystem for Computations with Non-commutative Polynomial Algebras. Centre for Computer Algebra, TU Kaiserslautern (2010). http://www.singular.uni-kl.de
Greuel, G.M., Pfister, G.: A SINGULAR Introduction to Commutative Algebra. Springer (2002)
Hong, H., Liska, R., Steinberg, S.: Testing stability by quantifier elimination. J. Symbolic Comput. 24(2), 161–187 (1997). http://www.sciencedirect.com/science/journal/07477171
La Scala, R., Levandovskyy, V.: Letterplace ideals and non-commutative Gröbner bases. J. Symbolic Comput. 44(10), 1374–1393 (2009). DOI doi:10.1016/j.jsc.2009.03. 002
Levandovskyy, V.: Non-commutative computer algebra for polynomial algebras: Gröbner bases, applications and implementation. Ph.D. Thesis, Universität Kaiserslautern (2005). http://kluedo.ub.uni-kl.de/volltexte/2005/1883/
Levin, A.: Difference algebra. Algebra and Applications. Springer, New York (2008)
Liska, R., Drska, L.: FIDE: a REDUCE package for automation of FInite difference method for solving pDE. In: Watanabe, S., Nagata M. (eds.) Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC’90), pp. 169–176. ACM Press and Addison-Wesley (1990). http://www.acm.org:80/pubs/citations/proceedings/issac/96877/p169-liska/
Pommaret, J.F.: Partial differential control theory. Vol. 1: Mathematical tools. Vol. 2: Control systems. Mathematics and its Applications (Dordrecht) 530. Dordrecht: Kluwer Academic Publishers (2001)
Ritt, J.F.: Differential algebra. American Mathematical Society (AMS) (1950)
Saito, S., Sturmfels, B., Takayama, N.: Gröbner Deformations of Hypergeometric Differential Equations. Springer (2000)
Seiler, W.M.: Involution. The formal theory of differential equations and its applications in computer algebra. Algorithms and Computation in Mathematics 24. Springer, Berlin (2010). DOI 10.1007/978-3-642-01287-7
Thomas, J.: Numerical partial differential equations: Finite difference methods. Springer (1995)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag/Wien
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-7091-0794-2_7
Published:
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-0793-5
Online ISBN: 978-3-7091-0794-2
eBook Packages: Computer ScienceComputer Science (R0)