Computation of the Strength of PDEs of Mathematical Physics and their Difference Approximations

TL;DR

This paper introduces a novel method for evaluating the strength of PDEs and their difference approximations using Hilbert-type dimension polynomials, with algorithms based on Gr"obner bases, applied to fundamental physics equations.

Contribution

It develops a new computational technique for assessing the strength of PDEs and difference schemes, enabling comparison of numerical methods in mathematical physics.

Findings

Determined the strength of Maxwell and diffusion equations.

Compared the strength of forward and symmetric difference schemes.

Provided algorithms for computing Hilbert-type dimension polynomials.

Abstract

We develop a method for evaluation of A. Einstein's strength of systems of partial differential and difference equations based on the computation of Hilbert-type dimension polynomials of the associated differential and difference field extensions. Also we present algorithms for such computations, which are based on the Gr\"obner basis method adjusted for the modules over rings of differential, difference and inversive difference operators. The developed technique is applied to some fundamental systems of PDEs of mathematical physics such as the diffusion equation, Maxwell equations and equations for an electromagnetic field given by its potential. In each of these cases we determine the strength of the original system of PDEs and the strength of the corresponding systems of partial difference equations obtained by forward and symmetric difference schemes. In particular, we obtain a method for comparing two difference schemes from the point of view of their strength.