A Strongly Consistent Finite Difference Scheme for Steady Stokes Flow and its Modified Equations

TL;DR

This paper presents a strongly consistent second-order finite difference scheme for steady 2D Stokes flow, explicitly incorporating the pressure Poisson equation, and analyzes its accuracy and consistency using advanced algebraic techniques.

Contribution

The authors develop a novel finite difference scheme for Stokes flow that ensures strong consistency and incorporates the pressure equation explicitly, using differential and difference algebra methods.

Findings

The scheme achieves second-order accuracy.

It demonstrates strong consistency with the Stokes equations.

Performance comparison with the marker-and-cell method shows competitive results.

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/Groebner 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.