On Consistency of Finite Difference Approximations to the Navier-Stokes Equations

TL;DR

This paper compares three finite difference methods for the 2D Navier-Stokes equations, demonstrating that only one is strongly consistent and performs better in numerical tests.

Contribution

It introduces a computer algebra-assisted method for generating finite difference approximations and proves its strong consistency with the Navier-Stokes equations.

Findings

Only one approximation is strongly consistent.

The consistent approximation shows better numerical behavior.

The other two approximations lack strong consistency.

Abstract

In the given paper, we confront three finite difference approximations to the Navier--Stokes equations for the two-dimensional viscous incomressible fluid flows. Two of these approximations were generated by the computer algebra assisted method proposed based on the finite volume method, numerical integration, and difference elimination. The third approximation was derived by the standard replacement of the temporal derivatives with the forward differences and the spatial derivatives with the central differences. We prove that only one of these approximations is strongly consistent with the Navier--Stokes equations and present our numerical tests which show that this approximation has a better behavior than the other two.