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Consistency Analysis of Finite Difference Approximations to PDE Systems

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Mathematical Modeling and Computational Science (MMCP 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7125))

Abstract

We consider finite difference approximations to systems of polynomially-nonlinear partial differential equations the coefficients of which are rational functions over rationals in the independent variables. The notion of strong consistency which we introduced earlier for linear systems is extended to nonlinear ones. For orthogonal and uniform grids we describe an algorithmic procedure for the verification of the strong consistency based on the computation of difference standard bases. The concepts and algorithmic methods of the present paper are illustrated by two finite difference approximations to the two-dimensional Navier-Stokes equations. One of these approximations is strongly consistent, while the other is not.

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References

  1. Bächler, T., Gerdt, V.P., Lange-Hegermann, M., Robertz, D.: Thomas Decomposition of Algebraic and Differential Systems. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2010. LNCS, vol. 6244, pp. 31–54. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  2. Becker, T., Weispfenning, V.: Gröbner Bases: A Computational Approach to Commutative Algebra. Graduate Texts in Mathematics, vol. 141. Springer, New York (1993)

    MATH  Google Scholar 

  3. Blinkov, Y.A., Cid, C.F., Gerdt, V.P., Plesken, W., Robertz, D.: The MAPLE Package Janet: II. Linear Partial Differential Equations. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Proceedings of the 6th International Workshop on Computer Algebra in Scientific Computing, pp. 41–54. Technische Universität München (2003), http://wwwb.math.rwth-aachen.de/Janet

  4. Cox, D., Little, J., O’Shie, D.: Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edn. Springer, New York (2007)

    Book  Google Scholar 

  5. Dorodnitsyn, V.: The Group Properties of Difference Equations. Moscow, Fizmatlit (2001) (in Russian)

    Google Scholar 

  6. Gerdt, V.P.: Completion of Linear Differential Systems to Involution. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 1999. Computer Algebra in Scientific Computing / CASC 1999, pp. 115–137. Springer, Berlin (1999) arXiv:math.AP/9909114

    Google Scholar 

  7. Gerdt, V.P.: Involutive Algorithms for Computing Gröbner Bases. In: Cojocaru, S., Pfister, G., Ufnarovsky, V. (eds.) Computational Commutative and Non-Commutative Algebraic Geometry, pp. 199–225. IOS Press, Amsterdam (2005) arXiv:math.AC/0501111

    Google Scholar 

  8. Gerdt, V.P.: On Decomposition of Algebraic PDE Systems into Simple Subsystems. Acta Appl. Math. 101, 39–51 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gerdt, V.P., Blinkov, Y.A.: Involutive Bases of Polynomial Ideals. Math. Comput. Simulat. 45, 519–542 (1998) arXiv:math.AC/9912027

    Article  MathSciNet  MATH  Google Scholar 

  10. Gerdt, V.P., Blinkov, Y.A.: Involution and Difference Schemes for the Navier–Stokes Equations. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 94–105. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  11. Gerdt, V.P., Blinkov, Y.A., Mozzhilkin, V.V.: Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations. SIGMA 2, 051 (2006) arXiv:math.RA/0605334

    MATH  Google Scholar 

  12. Gerdt, V.P., Robertz, D.: A Maple Package for Computing Gröbner Bases for Linear Recurrence Relations. Nucl. Instrum. Methods 559(1), 215–219 (2006) arXiv:cs.SC/0509070, http://wwwb.math.rwth-aachen.de/Janet

    Article  Google Scholar 

  13. Gerdt, V.P., Robertz, D.: Consistency of Finite Difference Approximations for Linear PDE Systems and its Algorithmic Verification. In: Watt, S.M. (ed.) Proceedings of ISSAC 2010, pp. 53–59. Association for Computing Machinery (2010)

    Google Scholar 

  14. Hubert, E.: Notes on Triangular Sets and Triangulation-Decomposition Algorithms II: Differential Systems. In: Winkler, F., Langer, U. (eds.) SNSC 2001. LNCS, vol. 2630, pp. 40–87. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  15. Gresho, P.M., Sani, R.L.: On Pressure Boundary Conditions for the Incompressible Navier-Stokes Equations. Int. J. Numer. Meth. Fl. 7, 1111–1145 (1987)

    Article  MATH  Google Scholar 

  16. Janet, M.: Leçons sur les Systèmes d’Equations aux Dérivées Partielles. Cahiers Scientifiques, IV. Gauthier-Villars, Paris (1929)

    Google Scholar 

  17. Kim, J., Moin, P.: Application of a Fractional-Step Method To Imcompressible Navier-Stokes Equations. J. Comput. Phys. 59, 308–323 (1985)

    Article  MATH  Google Scholar 

  18. La Scala, R., Levandovskyy, V.: Skew Polynomila Rings, Gröbner Bases and The Letterplace Embedding of the Free Associative Algebra, arXiv:math.RA/0230289

    Google Scholar 

  19. Levin, A.: Difference Algebra. Algebra and Applications, vol. 8. Springer, Heidelberg (2008)

    Book  MATH  Google Scholar 

  20. Martin, B., Levandovskyy, V.: 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: Progress and Prospects, pp. 123–156. Springer, Wien (2012)

    Google Scholar 

  21. Ollivier, F.: Standard Bases of Differential Ideals. In: Sakata, S. (ed.) AAECC 1990. LNCS, vol. 508, pp. 304–321. Springer, Heidelberg (1991)

    Chapter  Google Scholar 

  22. Pozrikidis, C.: Fluid Dynamics: Theory, Computation and Numerical Simulation. Kluwer, Amsterdam (2001)

    Book  MATH  Google Scholar 

  23. Rosinger, E.E.: Nonlinear Equivalence, Reduction of PDEs to ODEs and Fast Convergent Numerical Methods. Pitman, London (1983)

    MATH  Google Scholar 

  24. Samarskii, A.A.: Theory of Difference Schemes. Marcel Dekker, New York (2001)

    Book  MATH  Google Scholar 

  25. Seiler, W.M.: Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra. In: Algorithms and Computation in Mathematics, vol. 24. Springer, Heidelberg (2010)

    Google Scholar 

  26. Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations, 2nd edn. SIAM, Philadelphia (2004)

    Book  MATH  Google Scholar 

  27. Thomas, J.M.: Differential Systems. AMS Colloquium Publications XX1 (1937); Systems and Roots. The Wylliam Byrd Press, Rychmond, Virginia (1962)

    Google Scholar 

  28. Thomas, J.W.: Numerical Partial Differential Equations: Finite Difference Methods, 2nd edn. Springer, New York (1998)

    Google Scholar 

  29. Thomas, J.W.: Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations. Springer, New York (1999)

    Book  MATH  Google Scholar 

  30. Trushin, D.V.: Difference Nullstellensatz, arXiv:math.AC/0908.3865

    Google Scholar 

  31. Zobnin, A.: Admissible Orderings and Finiteness Criteria for Differential Standard Bases. In: Kauers, M. (ed.) Proceedings of ISSAC 2005, pp. 365–372. Association for Computing Machinery (2010)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute for Nuclear Research, 141980, Dubna, Russia

    Vladimir P. Gerdt

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  1. Vladimir P. Gerdt
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Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute for Nuclear Research, 6 Joliot Curie St., 141980, Dubna, Russia

    Gheorghe Adam

  2. Technical University of Košice, Němcovej 32, 042 00, Košice, Slovakia

    Ján Buša

  3. Pavol Jozef Šafárik University, Jesenná 5, 040 01, Košice, Slovakia

    Michal Hnatič

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Gerdt, V.P. (2012). Consistency Analysis of Finite Difference Approximations to PDE Systems. In: Adam, G., Buša, J., Hnatič, M. (eds) Mathematical Modeling and Computational Science. MMCP 2011. Lecture Notes in Computer Science, vol 7125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28212-6_3

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  • DOI: https://doi.org/10.1007/978-3-642-28212-6_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-28211-9

  • Online ISBN: 978-3-642-28212-6

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