A Strongly Consistent Finite Difference Scheme for Steady Stokes Flow and its Modified Equations
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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.
KeywordsComputer algebra Difference elimination Finite difference approximation Janet basis Modified equations Stokes flow Strong consistency
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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