A polyhedral discrete de Rham numerical scheme for the Yang-Mills equations

TL;DR

This paper introduces a novel discrete de Rham scheme for Yang-Mills equations that exactly preserves the non-linear constraint, enabling accurate and stable 3D numerical simulations.

Contribution

It develops a Lie algebra-valued discrete de Rham scheme that maintains the non-linear constraint of Yang-Mills equations exactly, with proven energy estimates and numerical validation.

Findings

Exact preservation of the non-linear constraint in discretisation.

Energy estimates demonstrating scheme stability.

Successful 3D numerical simulations confirming effectiveness.

Abstract

We present a discretisation of the 3+1 formulation of the Yang-Mills equations in the temporal gauge, using a Lie algebra-valued extension of the discrete de Rham (DDR) sequence, that preserves the non-linear constraint exactly. In contrast to Maxwell's equations, where the preservation of the analogous constraint only depends on reproducing some complex properties of the continuous de Rham sequence, the preservation of the non-linear constraint relies for the Yang-Mills equations on a constrained formulation, previously proposed in [10]. The fully discrete nature of the DDR method requires to devise appropriate constructions of the non-linear terms, adapted to the discrete spaces and to the need for replicating the crucial Ad-invariance property of the $L^2$-product. We then prove some energy estimates, and provide results of 3D numerical simulations based on this scheme.