From Exact Space-Time Symmetry Conservation to Automatic Mesh Refinement in Discrete Initial Boundary Value Problems

TL;DR

This paper introduces a variational approach to IBVPs that preserves space-time symmetries and Noether charges after discretization, enabling automatic mesh refinement, demonstrated on scalar wave propagation.

Contribution

It develops a novel variational formulation for IBVPs that maintains symmetries and conservation laws discretely, allowing for adaptive mesh refinement based on Noether charge conservation.

Findings

Exact conservation of Noether charges in discrete IBVPs.

Automatic mesh refinement guided by symmetry preservation.

Applicability to high-order discretizations with SBP operators.

Abstract

In this contribution we present recent developments in the formulation and solution of Initial Boundary Value Problems (IBVPs). Building upon a modern variational action formulation of classical dynamics, we treat Initial Boundary Value Problems directly on the action level, bypassing governing equations. We show that by including coordinate maps as dynamical degrees of freedom together with propagating fields two key results emerge. Space-time symmetries remain protected even after discretization, leading to an exact conservation of Noether charges even for discrete IBVPs. The dynamical nature of the coordinate maps leads to an adjustment of space-time resolution, guided by Noether charge conservation, realizing a form of automatic adaptive mesh refinement. We stress that as long as SBP operators are used for the discretization, our results are independent of whether the dynamics are solved on the action or governing equation level and hold in particular also at high order. As proof-of-principle for our approach we present its application to scalar wave-propagation in 1+1 dimensions.