A remedy for constraint growth in Numerical Relativity

TL;DR

This paper proposes adding spatial derivatives of constraints to evolution equations in numerical relativity, damping high-frequency violations and improving stability in simulations of highly gravitating systems.

Contribution

It introduces a novel method of constraint damping by incorporating spatial derivatives, applied to Maxwell's equations, with potential benefits for nonlinear Einstein equations.

Findings

High-frequency constraint violations are effectively damped.

Mixed hyperbolic-parabolic systems improve stability.

Constraint-preserving boundary conditions are successfully implemented.

Abstract

Rapid growth of constraints is often observed in free evolutions of highly gravitating systems. To alleviate this problem we investigate the effect of adding spatial derivatives of the constraints to the right hand side of the evolution equations, and we look at how this affects the character of the system and the treatment of boundaries. We apply this technique to two formulations of Maxwell's equations, the so-called fat Maxwell and the Knapp-Walker-Baumgarte systems, and obtain mixed hyperbolic-parabolic problems in which high frequency constraint violations are damped. Constraint-preserving boundary conditions amount to imposing Dirichlet boundary conditions on constraint variables, which translate into Neumann-like boundary conditions for the main variables. The success of the numerical tests presented in this work suggests that this remedy may bring benefits to fully nonlinear simulations of General Relativity.