Algorithmic approach to strong consistency analysis of finite difference approximations to PDE systems
For researchers in numerical PDEs, this provides a systematic verification tool for strong consistency, though the method is limited to Cartesian grids and specific nonlinearities.
This paper presents an algorithmic method for verifying strong consistency of finite difference approximations for a class of polynomially nonlinear PDE systems. The approach is applied to the incompressible Navier-Stokes equations, yielding s-consistent difference schemes including the pressure Poisson equation.
For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach to the s(trong)-consistency analysis of their finite difference approximations on Cartesian grids. First we apply the differential Thomas decomposition to the input system, resulting in a partition of the solution set. We consider the output simple subsystem that contains a solution of interest. Then, for this subsystem, we suggest an algorithm for verification of s-consistency for its finite difference approximation. For this purpose we develop a difference analogue of the differential Thomas decomposition, both of which jointly allow to verify the s-consistency of the approximation. As an application of our approach, we show how to produce s-consistent difference approximations to the incompressible Navier-Stokes equations including the pressure Poisson equation.