A symmetry and Noether charge preserving discretization of initial value problems

TL;DR

This paper introduces a geometric variational discretization method for second order initial value problems that preserves symmetries and Noether charges by discretizing along a world-line parameter, leading to adaptive mesh refinement.

Contribution

The authors develop a novel discretization approach inspired by general relativity that maintains translation symmetry and conserved quantities in numerical simulations.

Findings

Exact preservation of Noether charge in simulations

Automatic adaptive mesh refinement guided by symmetries

Demonstrated convergence with explicit examples

Abstract

Taking insight from the theory of general relativity, where space and time are treated on the same footing, we develop a novel geometric variational discretization for second order initial value problems (IVPs). By discretizing the dynamics along a world-line parameter, instead of physical time directly, we retain manifest translation symmetry and conservation of the associated continuum Noether charge. A non-equidistant time discretization emerges dynamically, realizing a form of automatic adaptive mesh refinement (AMR), guided by the system symmetries. Using appropriately regularized summation by parts finite difference operators, the continuum Noether charge, defined via the Killing vector associated with translation symmetry, is shown to be exactly preserved in the interior of the simulated time interval. The convergence properties of the approach are demonstrated with two explicit examples.