A Symbolic Approach to Generation and Analysis of Finite Difference Schemes of Partial Differential Equations

TL;DR

This paper introduces three symbolic methods for generating finite difference schemes for PDEs, proves their equivalence for linear constant-coefficient PDEs, and applies algebraic techniques to analyze stability and dispersion.

Contribution

It presents a unified symbolic framework for scheme generation and stability analysis, including implementation in SINGULAR and Mathematica for linear PDEs.

Findings

Three symbolic approaches are equivalent for linear PDEs with constant coefficients.

The use of cylindrical algebraic decomposition helps derive stability conditions.

Implementation demonstrates effective scheme generation and stability analysis.

Abstract

In this paper we discuss three symbolic approaches for the generation of a finite difference scheme of a partial differential equation (PDE). We prove, that for a linear PDE with constant coefficients these three approaches are equivalent and discuss the applicability of them to nonlinear PDE's as well as to the case of variable coefficients. Moreover, we systematically use another symbolic technique, namely the cylindrical algebraic decomposition, in order to derive the conditions on the von Neumann stability of a difference s cheme for a linear PDE with constant coefficients. For stable schemes we demonst rate algorithmic and symbolic approach to handle both continuous and discrete di spersion. We present an implementation of tools for generation of schemes, which rely on Gr\"obner basis, in the system SINGULAR and present numerous e xamples, computed with our implementation. In the stability analysis, we use the system MATHEMATICA for cylindrical algebraic decomposition.