Invariant meshless discretization schemes

TL;DR

This paper introduces a novel meshless discretization method that preserves Lie symmetries of differential equations, leading to improved numerical solutions compared to non-invariant schemes.

Contribution

The paper presents a new invariant meshless discretization scheme utilizing equivariant moving frames to preserve symmetries in numerical approximations.

Findings

Invariant schemes outperform non-invariant schemes in numerical accuracy.

The method effectively constructs invariant approximations for nonlinear diffusion equations.

Numerical tests demonstrate the superiority of invariant meshless schemes.

Abstract

A method is introduced for the construction of meshless discretization schemes which preserve Lie symmetries of the differential equations that these schemes approximate. The method exploits the fact that equivariant moving frames provide a way of associating invariant functions to non-invariant functions. An invariant meshless approximation of a nonlinear diffusion equation is constructed. Comparative numerical tests with a non-invariant meshless scheme are presented. These tests yield that invariant meshless schemes can lead to substantially improved numerical solutions compared to numerical solutions generated by non-invariant meshless schemes.