Exact symmetry conservation and automatic mesh refinement in discrete initial boundary value problems

TL;DR

This paper introduces a symmetry-preserving, reparametrization-invariant discretization method for initial boundary value problems that enables exact conservation laws and automatic mesh refinement, demonstrated through a 1+1 dimensional wave propagation example.

Contribution

It develops a novel action-based discretization approach incorporating coordinate maps as dynamical variables, ensuring symmetry preservation and enabling automatic adaptive mesh refinement.

Findings

Exact conservation of Noether charges after discretization

Automatic mesh refinement driven by symmetry considerations

Successful numerical demonstration with wave propagation in 1+1 dimensions

Abstract

We present a novel solution procedure for initial boundary value problems. The procedure is based on an action principle, in which coordinate maps are included as dynamical degrees of freedom. This reparametrization invariant action is formulated in an abstract parameter space and an energy density scale associated with the space-time coordinates separates the dynamics of the coordinate maps and of the propagating fields. Treating coordinates as dependent, i.e. dynamical quantities, offers the opportunity to discretize the action while retaining all space-time symmetries and also provides the basis for automatic adaptive mesh refinement (AMR). The presence of unbroken space-time symmetries after discretization also ensures that the associated continuum Noether charges remain exactly conserved. The presence of coordinate maps in addition provides new freedom in the choice of boundary conditions. An explicit numerical example for wave propagation in $1+1$ dimensions is provided, using recently developed regularized summation-by-parts finite difference operators.