Abstract
We construct and analyze a strongly consistent second-order finite difference scheme for the steady two-dimensional Stokes flow. The pressure Poisson equation is explicitly incorporated into the scheme. Our approach suggested by the first two authors is based on a combination of the finite volume method, difference elimination, and numerical integration. We make use of the techniques of the differential and difference Janet/Gröbner bases. In order to prove strong consistency of the generated scheme we correlate the differential ideal generated by the polynomials in the Stokes equations with the difference ideal generated by the polynomials in the constructed difference scheme. Additionally, we compute the modified differential system of the obtained scheme and analyze the scheme’s accuracy and strong consistency by considering this system. An evaluation of our scheme against the established marker-and-cell method is carried out.
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References
Adams, W.W., Loustanau, P.: Introduction to Gröbner Bases. Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence (1994)
Amodio, P., Blinkov, Y., Gerdt, V., La Scala, R.: On consistency of finite difference approximations to the Navier-Stokes equations. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 46–60. Springer, Cham (2013). https://doi.org/10.1007/978-3-319-02297-0_4
Amodio, P., Blinkov, Y.A., Gerdt, V.P., La Scala, R.: 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)
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 6th International Workshop on Computer Algebra in Scientific Computing, CASC 2003, pp. 41–54. Technische Universität München (2003). Package Janet is freely available on the web pagehttp://wwwb.math.rwth-aachen.de/Janet/
Fancher, G.H., Lewis, J.A.: Flow of simple fluids through porous materials. Indus. Eng. Chem. Res. 25(10), 1139–1147 (1933)
Ganzha, V.G., Vorozhtsov, E.V.: Computer-Aided Analysis of Difference Schemes for Partial Differential Equations. Wiley, New York (1996)
Gerdt, V.P., Blinkov, Y.A., Mozzhilkin, V.V.: Gröbner bases and generation of difference schemes for partial differential equations. SIGMA 2, 051 (2006)
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). https://doi.org/10.1007/978-3-642-04103-7_10
Gerdt, V.P.: Consistency analysis of finite difference approximations to PDE systems. In: Adam, G., Buša, J., Hnatič, M. (eds.) MMCP 2011. LNCS, vol. 7125, pp. 28–42. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-28212-6_3
Gerdt, V.P.: Involutive algorithms for computing Gröbner bases. In: Cojocaru, S., Pfister, G., Ufnarovski, V. (eds.) Computational Commutative and Non-Commutative Algebraic Geometry, NATO Science Series, pp. 199–225. IOS Press (2005)
Gerdt, V.P., Robertz, D.: Computation of difference Gröbner bases. Comput. Sci. J. Moldova 20 2(59), 203–226 (2012). Package LDA is freely available on the web page http://wwwb.math.rwth-aachen.de/Janet/
Gerdt, V.P., Robertz, D.: Consistency of finite difference approximations for linear PDE systems and its algorithmic verification. In: Watt, S.M. (ed.) ISSAC 2010, pp. 53–59. Association for Computing Machinery, New York (2010)
Harlow, F.H., Welch, J.E.: Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface. Phys. Fluids 8, 2182–2189 (1965)
Kohr, M., Pop, I.: Viscous Incompressible Flow for Low Reynolds Numbers. Advances in Boundary Elements, vol. 16. WIT Press, Sauthampton (2004)
Levin, A.: Difference Algebra. Algebra and Applications, vol. 8. Springer, Heidelberg (2008). https://doi.org/10.1007/978-1-4020-6947-5
Milne-Tompson, L.M.: Theoretical Hydrodynamics, 5th edn. Macmillan Education LTD, Banjul (1968)
Moin, P.: Fundamentals of Engineering Numerical Analysis, 2nd edn. Cambridge University Press, Cambridge (2010)
Petersson, N.A.: Stability of pressure boundary conditions for Stokes and Navier-Stokes equations. J. Comput. Phys. 172, 40–70 (2001)
Seiler, W.M.: Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms and Computation in Mathematics, vol. 24. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-01287-7
Shokin, Y.I.: The Method of Differential Approximation. Springer, Berlin (1983)
Acknowledgments
The authors are grateful to Daniel Robertz for his help with respect to the use of the packages Janet and LDA and to the anonymous referees for their suggestions. This work has been partially supported by the King Abdullah University of Science and Technology (KAUST baseline funding), the Russian Foundation for Basic Research (16-01-00080) and the RUDN University Program (5-100).
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Blinkov, Y.A., Gerdt, V.P., Lyakhov, D.A., Michels, D.L. (2018). A Strongly Consistent Finite Difference Scheme for Steady Stokes Flow and its Modified Equations. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2018. Lecture Notes in Computer Science(), vol 11077. Springer, Cham. https://doi.org/10.1007/978-3-319-99639-4_5
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