Dimension Polynomials and the Einstein's Strength of Some Systems of Quasi-linear Algebraic Difference Equations

TL;DR

This paper develops a method using characteristic sets to analyze the algebraic structure of quasi-linear difference equations and evaluates the Einstein's strength of various finite-difference schemes for PDEs.

Contribution

It introduces a characteristic set method for inversive difference polynomials and applies it to assess the Einstein's strength of difference schemes for PDEs.

Findings

Computed difference dimension polynomials for specific systems.

Determined Einstein's strength of finite-difference schemes for PDEs.

Provided a comparative analysis of scheme effectiveness.

Abstract

In this paper we present a method of characteristic sets for inversive difference polynomials and apply it to the analysis of systems of quasi-linear algebraic difference equations. We describe characteristic sets and compute difference dimension polynomials associated with some such systems. Then we apply our results to the comparative analysis of difference schemes for some PDEs from the point of view of their Einstein's strength. In particular, we determine the Einstein's strength of standard finite-difference schemes for the Murray, Burgers and some other reaction-diffusion equations.