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Construction of a New Implicit Difference Scheme for 2D Boussinesq Paradigm Equation

  • Yu. A. Blinkov
  • V. P. GerdtEmail author
  • I. A. Pankratov
  • E. A. Kotkova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)

Abstract

Given an orthogonal and uniform solution grid with equal spatial grid sizes, we construct a new second-order implicit conservative finite difference scheme for the fourth-order 2D Boussinesq paradigm equation with quadratic nonlinear part. We apply the algebraic approach to the construction of difference schemes suggested by the first two authors and based on a combination of the finite volume method, difference elimination, and numerical integration. For the difference elimination, we make use of the techniques of Gröbner bases; in so doing, we introduce an extra difference indeterminate to reduce the nonlinear elimination problem to the pure linear one. It allows us to apply the Gröbner bases algorithm and software designed for linear generating sets of difference polynomials. Additionally, for the obtained difference scheme and also for another scheme known in the literature, we compute the modified differential equations and compare them.

Keywords

Computer algebra Difference elimination Finite difference approximation Gröbner basis Modified equation Consistency Conservativity 

Notes

Acknowledgements

This work has been partially supported by the Russian Foundation for Basic Research (grant No. 18-51-18005) and by the RUDN University Program (5-100).

References

  1. 1.
    Gerdt, V.P., Blinkov, Y.A., Mozzhilkin, V.V.: Gröbner bases and generation of difference schemes for partial differential equations. SIGMA 2, 051 (2006)zbMATHGoogle Scholar
  2. 2.
    Christov, C.I.: An energy - consistent Galilean - invariant dispersive shallow - water model. Wave Motion 34, 161–174 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Christov, C.I., Kolkovska, N., Vasileva, D.: On the numerical simulation of unsteady solutions for the 2D Boussinesq paradigm equation. In: Dimov, I., Dimova, S., Kolkovska, N. (eds.) NMA 2010. LNCS, vol. 6046, pp. 386–394. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-18466-6_46CrossRefzbMATHGoogle Scholar
  4. 4.
    Todorov, M.D.: Nonlinear Waves: Theory, Computer Simulation, Experiment. Morgan & Claypool Publishers, San Rafael (2018)zbMATHGoogle Scholar
  5. 5.
    Koren, B., Abgrall, R., Bochev, P., Frank, J.: Physics-compatible numerical methods. J. Comput. Phys. 257, 1039–1526 (2014)CrossRefGoogle Scholar
  6. 6.
    Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations, 2nd edn. SIAM, Philadelphia (2004)zbMATHGoogle Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    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_3CrossRefGoogle Scholar
  9. 9.
    Levin, A.: Difference Algebra. Algebra and Applications, vol. 8. Springer, Dordrecht (2008).  https://doi.org/10.1007/978-1-4020-6947-5CrossRefzbMATHGoogle Scholar
  10. 10.
    Gerdt, V., La Scala, R.: Noetherian quotients of the algebra of partial difference polynomials and Gröbner bases of symmetric ideals. J. Algebra 423, 1233–1261 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
  12. 12.
    Buchberger, B.: Gröbner bases: an algorithmic method in polynomial ideal theory. In: Bose, N.K. (ed.) Recent Trends in Multidimensional System, pp. 184–232. Reidel, Dordrecht (1985)CrossRefGoogle Scholar
  13. 13.
    Adams, W.W., Loustanau, P.: Introduction to Gröbner Bases. Graduate Studies in Mathematics, vol. 3. American Mathematical Society (1994)Google Scholar
  14. 14.
    Robertz, D.: Formal Algorithmic Elimination for PDEs. LNM, vol. 2121. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-11445-3CrossRefzbMATHGoogle Scholar
  15. 15.
    Chertock, A., Christov, C.I., Kurganov, A: Central-upwind schemes for Boussinesq paradigm equations. In: Krause, E., Shokin, Yu., Resch, M., Kröner, D., Shokina, N. (eds.) Computational Science and High Performance Computing IV, vol. 115, pp. 267–281 (2011)Google Scholar
  16. 16.
    Shokin, Y.I.: The Method of Differential Approximation, 1st edn. Springer, Heidelberg (1983)CrossRefGoogle Scholar
  17. 17.
    Ganzha, V.G., Vorozhtsov, E.V.: Computer-aided Analysis of Difference Schemes for Partial Differential Equations. Wiley, New York (1996)CrossRefGoogle Scholar
  18. 18.
    Moin, P.: Fundamentals of Engineering Numerical Analysis, 2nd edn. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  19. 19.
    Seiler, W.M.: Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms and Computation in Mathematics, vol. 24, 1st edn. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-01287-7CrossRefzbMATHGoogle Scholar
  20. 20.
    Blinkov, Yu.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, CASC 2003, pp. 41–54. Technische Universität München (2003). Package Janet is freely available on the web page http://134.130.169.213/Janet/
  21. 21.
    Gerdt, V.P., Robertz, D.: Computation of difference Gröbner bases. Comput. Sc. J. Moldova. 20(2), 203–226 (2012). Package LDA is freely available on the web page http://134.130.169.213/Janet/zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Yu. A. Blinkov
    • 1
  • V. P. Gerdt
    • 2
    • 3
    • 4
    Email author
  • I. A. Pankratov
    • 1
  • E. A. Kotkova
    • 2
    • 4
  1. 1.Saratov State UniversitySaratovRussian Federation
  2. 2.Joint Institute for Nuclear ResearchDubnaRussian Federation
  3. 3.Peoples’ Friendship University of RussiaMoscowRussian Federation
  4. 4.Dubna State UniversityDubnaRussian Federation

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