Symmetry Preserving Numerical Schemes for Partial Differential Equations and their Numerical Tests

TL;DR

This paper develops symmetry-preserving finite difference schemes for PDEs using equivariant moving frames, demonstrating improved accuracy over standard methods through numerical tests on specific equations.

Contribution

It introduces a systematic approach to construct invariant numerical schemes for PDEs using equivariant moving frames, enhancing accuracy and symmetry preservation.

Findings

Invariant schemes outperform standard discretizations in accuracy.

Numerical tests confirm the effectiveness of symmetry-preserving methods.

Schemes successfully applied to heat and spherical Burgers equations.

Abstract

The method of equivariant moving frames on multi-space is used to construct symmetry preserving finite difference schemes of partial differential equations invariant under finite-dimensional symmetry groups. Invariant numerical schemes for a heat equation with a logarithmic source and the spherical Burgers equation are obtained. Numerical tests show how invariant schemes can be more accurate than standard discretizations on uniform rectangular meshes.