Finite difference schemes are widely used in applied mathematics to numerically solve partial differential equations. However, for a given solution scheme, it is usually difficult to evaluate the quality of the underlying finite difference approximation with respect to the inheritance of algebraic properties of the differential problem under consideration. In this paper, we present an appropriate quality criterion of strong consistency for finite difference approximations to systems of nonlinear partial differential equations. This property strengthens the standard requirement of consistency of difference equations with differential ones. We use a verification algorithm for strong consistency, which is based on the computation of difference Gröbner bases. This allows for the evaluation and construction of solution schemes that preserve some fundamental algebraic properties of the system at the discrete level. We demonstrate the suggested approach by simulating a Kármán vortex street for the two-dimensional incompressible viscous flow described by the Navier–Stokes equations.
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P. Amodio, Yu. A. Blinkov, V. P. Gerdt, and R. La Scala, “On consistency of finite difference approximations to the Navier–Stokes equations,” in: V. P. Gerdt, W. Koepff, E. W. Mayr, and E. V. Vorozhtsov (eds.), Computer Algebra in Scientific Computing / CASC 2013, Lecture Notes Comput. Sci., 8136, Springer, Cham (2013), pp. 46–60.
P. Amodio, Yu. A. Blinkov, V. P. Gerdt, and R. La Scala, “Algebraic construction and numerical behavior of a new s-consistent difference scheme for the 2D Navier–Stokes equations,” Appl. Math. Comput., 314, 408–421 (2017).
T. Bächler, V. P. Gerdt, M. Lange-Hegermann, and D. Robertz, “Algorithmic Thomas decomposition of algebraic and differential systems,” J. Symbolic Comput., 47, No. 10, 1233–1266 (2012).
T. Becker and V. Weispfenning, A Computational Approach to Commutative Algebra, Springer, New York (1993).
Yu. A. Blinkov, C. F. Cid, V. P. Gerdt, W. Plesken, and D. Robertz, “The MAPLE package Janet: II. Linear partial differential equations,” in: V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov (eds.), Proceedings of the 6th International Workshop on Computer Algebra in Scientific Computing / CASC 2003, Technische Universit ät München (2003), pp. 41–54; the package Janet is freely available on the web page http://wwwb.math.rwth-aachen.de/Janet/.
D. Cox, J. Little, and D. O’Shie, Ideals, Varieties and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edition, Springer, New York (2007).
V. P. Gerdt, “On decomposition of algebraic PDE systems into simple subsystems,” Acta Appl. Math., 101, 39–51 (2008).
V. P. Gerdt, “Consistency analysis of finite difference approximations to PDE systems,” in: G. Adam, J. Buša, and M. Hnatič (eds.), Mathematical Modeling in Computational Physics / MMCP 2011, Lecture Notes Comput. Sci., 7125, Springer, Berlin (2012), pp. 28–42.
V. P. Gerdt, Yu. A. Blinkov, and V. V. Mozzhilkin, “Gröbner bases and generation of difference schemes for partial differential equations,” SIGMA, 2, 051 (2006).
V. P. Gerdt and Yu. A. Blinkov, “Involution and difference schemes for the Navier–Stokes equations,” in: V. P. Gerdt, E. W. Mayr, and E. V. Vorozhtsov (eds.), Computer Algebra in Scientific Computing / CASC 2009, Lecture Notes Comput. Sci., 5743, Springer, Berlin (2009), pp. 94–105.
V. P. Gerdt and R. La Scala, “Noetherian quotient of the algebra of partial difference polynomials and Gröbner bases of symmetric ideals, J. Algebra, 423, 1233–1261 (2015).
V. P. Gerdt and D. Robertz, “Consistency of finite difference approximations for linear PDE systems and its algorithmic verification,” in: S. M. Watt (ed.), Proceedings of ISSAC 2010, Association for Computing Machinery (2010), pp. 53–59.
V. P. Gerdt and D. Robertz, “Computation of difference Gröbner bases,” Comput. Sci. J. Moldova, 20, No. 2(59), 203–226 (2012).
S. K. Godunov and V. S. Ryabenkii, Difference Schemes. An Introduction to the Underlying Theory, Elsevier, New York (1987).
P. M. Gresho and R. L. Sani, “On pressure boundary conditions for the incompressible Navier–Stokes equations,” Internat. J. Numer. Methods Fluids, 7, 1111–1145 (1987).
E. Hubert, “Notes on triangular sets and triangulation-decomposition algorithms. II,” in: F. Winkler and U. Langer (eds.), Symbolic and Numerical Scientific Computation, Lecture Notes Comput. Sci., 2630, Springer, Berlin (2001), pp. 40–87.
T. Kármán, Aerodynamics, McGraw-Hill, New York (1963).
J. Kim and P. Moin, “Application of a fractional-step method to incompressible Navier–Stokes equations,” J. Comput. Phys., 59, 308–323 (1985).
E. R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York–London (1973).
M. Lange-Hegermann, “DifferentialThomas,” freely available on the web page https://wwwb.math.rwth-aachen.de/thomasdecomposition/.
A. Levin, Difference Algebra, Springer (2008).
K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations. An Introduction, Cambridge Univ. Press (2005).
F. Ollivier, “Standard bases of differential ideals,” in: S. Sakata (ed.), Applied Algebra. Algebraic Algorithms and Error-Correcting Codes/AAECC’90, Lecture Notes Comput. Sci., 508, Springer, London (1990), pp. 304–321.
D. Robertz, Formal Algorithmic Elimination for PDEs, Springer, Berlin (2014).
D. Robertz, LDA (Linear Difference Algebra), Aachen (2015), freely available on the web page http://134.130.169.213/Janet/.
A. A. Samarskii, Theory of Difference Schemes, Marcel Dekker, New York (2001).
W. M. Seiler, Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra, Springer, Heidelberg (2010).
Yu. I. Shokin, “On conservatism of difference schemes of gas dynamics,” in: F. G. Zhuang and Y. L. Zhu (eds.), Tenth International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes Physics, 264, Springer, Berlin (1986), pp. 578–583.
A. F. Sidorov, V. P. Shapeev, and N. N. Yanenko, Method of Differential Constraints and its Applications to Gas Dynamics [in Russian], Nauka, Novosibirsk (1984).
J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd edition, SIAM, Philadelphia (2004).
J. M. Thomas, Differential Systems, AMS Colloquium Publications, XXI (1937); Systems and Roots, The Wylliam Byrd Press, Rychmond, Virginia (1962).
J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, 2nd edition, Springer, New York (1998).
D. V. Trushin, Difference Nullstellensatz, arXiv:0908.3865[math.AC] (2011).
A. Zobnin, “Admissible orderings and finiteness criteria for differential standard bases,” in: M. Kauers (ed.), Proceedings of ISSAC 2005, Association for Computing Machinery (2010), pp. 365–372.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 468, 2018, pp. 249–266.
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Michels, D.L., Gerdt, V.P., Blinkov, Y.A. et al. On the Consistency Analysis of Finite Difference Approximations. J Math Sci 240, 665–677 (2019). https://doi.org/10.1007/s10958-019-04383-x
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DOI: https://doi.org/10.1007/s10958-019-04383-x