Illustrating Stability Properties of Numerical Relativity in Electrodynamics

TL;DR

This paper demonstrates that a reformulation of Maxwell's equations, inspired by stable formulations in general relativity, improves numerical stability and offers insights into the stability properties of electrodynamics and gravity.

Contribution

It introduces a reformulation of Maxwell's equations analogous to stable ADM formulations in general relativity, enhancing numerical stability and understanding.

Findings

Reformulated Maxwell's equations show improved numerical stability.

Numerical behavior reflects properties of linearized general relativity.

Framework may guide further numerical scheme improvements.

Abstract

We show that a reformulation of the ADM equations in general relativity, which has dramatically improved the stability properties of numerical implementations, has a direct analogue in classical electrodynamics. We numerically integrate both the original and the revised versions of Maxwell's equations, and show that their distinct numerical behavior reflects the properties found in linearized general relativity. Our results shed further light on the stability properties of general relativity, illustrate them in a very transparent context, and may provide a useful framework for further improvement of numerical schemes.