Consistency Analysis of Finite Difference Approximations to PDE Systems
This work provides a formal verification tool for finite difference schemes, relevant to researchers in numerical analysis and PDE simulation.
The paper extends the notion of strong consistency from linear to nonlinear PDE systems for finite difference approximations on orthogonal and uniform grids, and provides an algorithmic verification procedure using difference standard bases. The method is demonstrated on two approximations of the 2D Navier-Stokes equations, one strongly consistent and one not.
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.