Consistency Analysis of Finite Difference Approximations to PDE Systems

TL;DR

This paper extends the concept of strong consistency to nonlinear PDE systems and presents an algorithmic method using difference standard bases to verify consistency, demonstrated on Navier-Stokes equations.

Contribution

It introduces a new approach for verifying strong consistency of finite difference schemes for nonlinear PDEs using difference standard bases.

Findings

One scheme is strongly consistent with the PDE system.

Another scheme is not strongly consistent.

The method is illustrated with Navier-Stokes equations.

Abstract

In the given paper we consider finite difference approximations to systems of polynomially-nonlinear partial differential equations whose coefficients are rational functions over rationals in the independent variables. The notion of strong consistency which we introduced earlier for linear systems is extended to nonlinear ones. For orthogonal and uniform grids we describe an algorithmic procedure for verification of strong consistency based on computation of difference standard bases. The concepts and algorithmic methods of the present paper are illustrated by two finite difference approximations to the two-dimensional Navier-Stokes equations. One of these approximations is strongly consistent and another is not.