Energy Norms and the Stability of the Einstein Evolution Equations

TL;DR

This paper derives an energy norm-based exact expression for the growth rate of constraint violations in symmetric hyperbolic formulations of Einstein's equations, aiding in selecting stable formulations.

Contribution

It introduces an exact and an approximate algebraic method to estimate growth rates, improving the understanding of stability in Einstein evolution equations.

Findings

Exact growth rate expression matches numerical results.

Approximate rate estimate depends on initial data.

Method helps identify well-behaved formulations.

Abstract

The Einstein evolution equations may be written in a variety of equivalent analytical forms, but numerical solutions of these different formulations display a wide range of growth rates for constraint violations. For symmetric hyperbolic formulations of the equations, an exact expression for the growth rate is derived using an energy norm. This expression agrees with the growth rate determined by numerical solution of the equations. An approximate method for estimating the growth rate is also derived. This estimate can be evaluated algebraically from the initial data, and is shown to exhibit qualitatively the same dependence as the numerically-determined rate on the parameters that specify the formulation of the equations. This simple rate estimate therefore provides a useful tool for finding the most well-behaved forms of the evolution equations.