On the Consistency Analysis of Finite Difference Approximations

Abstract

Finite difference schemes are widely used in applied mathematics to numerically solve partial differential equations. However, for a given solution scheme, it is usually difficult to evaluate the quality of the underlying finite difference approximation with respect to the inheritance of algebraic properties of the differential problem under consideration. In this paper, we present an appropriate quality criterion of strong consistency for finite difference approximations to systems of nonlinear partial differential equations. This property strengthens the standard requirement of consistency of difference equations with differential ones. We use a verification algorithm for strong consistency, which is based on the computation of difference Gröbner bases. This allows for the evaluation and construction of solution schemes that preserve some fundamental algebraic properties of the system at the discrete level. We demonstrate the suggested approach by simulating a Kármán vortex street for the two-dimensional incompressible viscous flow described by the Navier–Stokes equations.