# Algorithmic approach to strong consistency analysis of finite difference approximations to PDE systems

## TL;DR

This paper presents an algorithmic method for analyzing the strong consistency of finite difference schemes for polynomially nonlinear PDE systems, using differential Thomas decomposition and its difference analogue.

## Contribution

It introduces a novel algorithm combining differential and difference Thomas decompositions to verify s-consistency of finite difference approximations for nonlinear PDEs.

## Key findings

- Verified s-consistency for Navier-Stokes equations
- Developed difference Thomas decomposition algorithm
- Provided a systematic approach for PDE approximation analysis

## Abstract

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.