On the Effect of Constraint Enforcement on the Quality of Numerical Solutions in General Relativity

TL;DR

This paper explores extending the equations of general relativity to better enforce constraints during numerical simulations, finding reduced constraint violations but no improvement in overall solution accuracy.

Contribution

It extends non-linear equations to improve constraint enforcement in numerical relativity, analyzing effects through numerical experiments.

Findings

Reduced growth of constraint violations

No improvement in solution accuracy

Constraint submanifold not an attractor for all solutions

Abstract

In Brodbeck et al 1999 it has been shown that the linearised time evolution equations of general relativity can be extended to a system whose solutions asymptotically approach solutions of the constraints. In this paper we extend the non-linear equations in similar ways and investigate the effect of various possibilities by numerical means. Although we were not able to make the constraint submanifold an attractor for all solutions of the extended system, we were able to significantly reduce the growth of the numerical violation of the constraints. Contrary to our expectations this improvement did not imply a numerical solution closer to the exact solution, and therefore did not improve the quality of the numerical solution.