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)


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.


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



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).


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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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