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

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/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-andcell method is carried out.

Algebraic Construction of a Strongly Consistent, Permutationally Symmetric and Conservative Difference Scheme for 3D Steady Stokes Flow

“…In this paper, 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. For the 3D steady Stokes flow, the governing system of partial differential equations (PDE) reads:…”

“…Here, x, y, z are the independent variables; the velocities u, v, and w, the pressure p, and the external forces f (1) , f (2) , and f (3) are the dependent variables; the constant Re is the Reynolds number; and ∆ := ∂ xx + ∂ yy + ∂ zz is the Laplace operator. Equation (1) approximate the incompressible Navier-Stokes equations:…”

“…) := p x + uu x + vu y + wu z − 1 Re ∆ u − f (1) = 0 , F (3) := p y + uv x + vv y + wv z − 1 Re ∆ v − f (2) = 0 , F (4) := p z + uw x + vw y + ww z − 1 Re ∆ w − f (3) = 0 .…”

“…In addition to the symmetry (3), among the algebraic properties of the PDE (1) to be preserved at the discrete level, we make sure of the conservativity and strong consistency or s-consistency of FDA. Conservativity means inheritance at the discrete level of the underlying conservation laws of PDE (1). The conventional notion of consistency (cf.…”

“…For Stokes flow, it means (see Definition 2) not only approximation of the equations in PDE by difference equations, but also approximation of elements in the differential ideal generated by the polynomials in (1) by elements in the difference ideal (cf. [8], Chapter 2) generated by the polynomials in the FDA to (1). In particular, if the s-consistency holds, then among the consequences of FDA, i.e., among elements in the ideal it generates, there are approximants of all integrability conditions (see [9], Chapter 2) for the original PDE.…”