On the consistency analysis of finite difference approximations

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 generally evaluate the quality of their underlying finite difference approximation with respect to the inheritance of algebraic properties of the differential problem under consideration. In this contribution, 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 the differential ones. On this foundation, 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, which preserve some fundamental algebraic properties of the system at the discrete level. We demonstrate our presented concept by simulating a Kármán vortex street for two-dimensional incompressible viscous flow described by the Navier–Stokes equations.