Invariant Discretization Schemes Using Evolution-Projection Techniques

TL;DR

This paper develops invariant discretization schemes for the heat equation using an evolution-projection approach that maintains symmetry properties and simplifies grid management, with proven convergence and numerical validation.

Contribution

It introduces a novel evolution-projection methodology for invariant schemes, enabling effective handling of moving grids while preserving Lie invariance.

Findings

Invariant schemes preserve symmetry properties.

Convergence rates are established and validated.

Numerical tests confirm the effectiveness of the approach.

Abstract

Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for non-invariant schemes of the heat equation in computational coordinates. We propose a new methodology for handling moving discretization grids which are generally indispensable for invariant numerical schemes. The idea is to use the invariant grid equation, which determines the locations of the grid point at the next time level only for a single integration step and then to project the obtained solution to the regular grid using invariant interpolation schemes. This guarantees that the scheme is invariant and allows one to work on the simpler stationary grids. The discretization errors of the invariant schemes are established and their convergence rates are estimated. Numerical tests are carried out to shed some light on the numerical properties of invariant discretization schemes using the proposed evolution-projection strategy.