Toward Making the Constraint Hypersurface an Attractor in Free Evolution

TL;DR

This paper proposes a general framework for modifying evolution equations to make the constraint hypersurface an attractor, demonstrated through Maxwell equations, aiming to improve the stability of numerical simulations.

Contribution

It introduces a universal method for adjusting evolution equations to enhance constraint preservation, applicable across various PDE systems.

Findings

Framework successfully applied to Maxwell equations

Potential to improve numerical relativity simulations

Enhances stability by making constraints attractors

Abstract

There is an abundance of empirical evidence in the numerical relativity literature that the form in which the Einstein evolution equations are written plays a significant role in the lifetime of numerical simulations. This paper attempts to present a consistent framework for modifying any system of evolution equations by adding terms that push the evolution toward the constraint hypersurface. The method is, in principle, applicable to any system of partial differential equations which can be divided into evolution equations and constraints, although it is only demonstrated here through an application to the Maxwell equations.