Consistency Analysis of Finite Difference Approximations to PDE Systems

  • Vladimir P. Gerdt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7125)


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.


systems of partial differential equations involution Thomas decomposition finite difference approximations consistency difference standard bases Navier-Stokes equations computer algebra 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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)CrossRefGoogle Scholar
  2. 2.
    Becker, T., Weispfenning, V.: Gröbner Bases: A Computational Approach to Commutative Algebra. Graduate Texts in Mathematics, vol. 141. Springer, New York (1993)zbMATHGoogle Scholar
  3. 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),
  4. 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)CrossRefGoogle Scholar
  5. 5.
    Dorodnitsyn, V.: The Group Properties of Difference Equations. Moscow, Fizmatlit (2001) (in Russian)Google Scholar
  6. 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/9909114Google Scholar
  7. 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/0501111Google Scholar
  8. 8.
    Gerdt, V.P.: On Decomposition of Algebraic PDE Systems into Simple Subsystems. Acta Appl. Math. 101, 39–51 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gerdt, V.P., Blinkov, Y.A.: Involutive Bases of Polynomial Ideals. Math. Comput. Simulat. 45, 519–542 (1998) arXiv:math.AC/9912027MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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)CrossRefGoogle Scholar
  11. 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/0605334zbMATHGoogle Scholar
  12. 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, CrossRefGoogle Scholar
  13. 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. 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)CrossRefGoogle Scholar
  15. 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)CrossRefzbMATHGoogle Scholar
  16. 16.
    Janet, M.: Leçons sur les Systèmes d’Equations aux Dérivées Partielles. Cahiers Scientifiques, IV. Gauthier-Villars, Paris (1929)Google Scholar
  17. 17.
    Kim, J., Moin, P.: Application of a Fractional-Step Method To Imcompressible Navier-Stokes Equations. J. Comput. Phys. 59, 308–323 (1985)CrossRefzbMATHGoogle Scholar
  18. 18.
    La Scala, R., Levandovskyy, V.: Skew Polynomila Rings, Gröbner Bases and The Letterplace Embedding of the Free Associative Algebra, arXiv:math.RA/0230289Google Scholar
  19. 19.
    Levin, A.: Difference Algebra. Algebra and Applications, vol. 8. Springer, Heidelberg (2008)CrossRefzbMATHGoogle Scholar
  20. 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. 21.
    Ollivier, F.: Standard Bases of Differential Ideals. In: Sakata, S. (ed.) AAECC 1990. LNCS, vol. 508, pp. 304–321. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  22. 22.
    Pozrikidis, C.: Fluid Dynamics: Theory, Computation and Numerical Simulation. Kluwer, Amsterdam (2001)CrossRefzbMATHGoogle Scholar
  23. 23.
    Rosinger, E.E.: Nonlinear Equivalence, Reduction of PDEs to ODEs and Fast Convergent Numerical Methods. Pitman, London (1983)zbMATHGoogle Scholar
  24. 24.
    Samarskii, A.A.: Theory of Difference Schemes. Marcel Dekker, New York (2001)CrossRefzbMATHGoogle Scholar
  25. 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. 26.
    Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations, 2nd edn. SIAM, Philadelphia (2004)CrossRefzbMATHGoogle Scholar
  27. 27.
    Thomas, J.M.: Differential Systems. AMS Colloquium Publications XX1 (1937); Systems and Roots. The Wylliam Byrd Press, Rychmond, Virginia (1962)Google Scholar
  28. 28.
    Thomas, J.W.: Numerical Partial Differential Equations: Finite Difference Methods, 2nd edn. Springer, New York (1998)Google Scholar
  29. 29.
    Thomas, J.W.: Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  30. 30.
    Trushin, D.V.: Difference Nullstellensatz, arXiv:math.AC/0908.3865Google Scholar
  31. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Vladimir P. Gerdt
    • 1
  1. 1.Laboratory of Information TechnologiesJoint Institute for Nuclear ResearchDubnaRussia

Personalised recommendations