Abstract
By using symbolic algebraic computation, we construct a strongly-consistent second-order finite difference scheme for steady three-dimensional Stokes flow and a Cartesian solution grid. The scheme has the second order of accuracy and incorporates the pressure Poisson equation. This equation is the integrability condition for the discrete momentum and continuity equations. Our algebraic approach to the construction of difference schemes suggested by the second and the third authors combines the finite volume method, numerical integration, and difference elimination. We make use of the techniques of the differential and difference Janet/Gröbner bases for performing related computations. To prove the strong consistency of the generated scheme, we use these bases to 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. As this takes place, our difference scheme is conservative and inherits permutation symmetry of the differential Stokes flow. For the obtained scheme, we compute the modified differential system and use it to analyze the scheme’s accuracy.
Highlights
We extend to the three-dimensional case the results of paper [1], where we generated and studied the strong consistent finite difference scheme for two-dimensional (2D) steady Stokes flow of an incompressible fluid
finite difference approximation (FDA), which we refer to as modified Stokes flow, and use it to analyze the order of approximation of the scheme and its consistency
Steady Stokes flow for the first time, a modified equation was constructed for non-evolutionary system of partial differential equations (PDE)
Summary
We extend to the three-dimensional case the results of paper [1], where we generated and studied the strong consistent finite difference scheme for two-dimensional (2D) steady Stokes flow of an incompressible fluid. The scheme contains a discrete version of the pressure Poisson equation (integrability condition) and is s-consistent with (1) For this purpose, we use the approach proposed in [10]. We apply the algorithmic criterion for verification of its s-consistency This criterion was designed in [6] and relates Janet/Gröbner bases of the differential and difference ideals generated by PDE and FDA. FDA, which we refer to as modified Stokes flow, and use it to analyze the order of approximation of the scheme and its consistency. PDE systems, and in [1], by applying the technique of differential Janet/Gröbner bases to the 2D steady Stokes flow for the first time, a modified equation was constructed for non-evolutionary system of PDE.
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