Diagonalizability of Constraint Propagation Matrices

TL;DR

This paper analyzes the eigenvalue properties of constraint propagation matrices in general relativity simulations, emphasizing the importance of diagonalizability for stability and providing criteria for different asymptotic behaviors of constraint violations.

Contribution

It introduces a classification of asymptotic behaviors of constraint violations and highlights the significance of matrix diagonalizability in ensuring stable constraint evolution.

Findings

Degeneracy of eigenvalues can cause divergence in constraint evolution.

Diagonalizability of matrices is crucial for stable constraint propagation.

The analysis applies broadly to numerical constrained dynamics.

Abstract

In order to obtain stable and accurate general relativistic simulations, re-formulations of the Einstein equations are necessary. In a series of our works, we have proposed using eigenvalue analysis of constraint propagation equations for evaluating violation behavior of constraints. In this article, we classify asymptotical behaviors of constraint-violation into three types (asymptotically constrained, asymptotically bounded, and diverge), and give their necessary and sufficient conditions. We find that degeneracy of eigenvalues sometimes leads constraint evolution to diverge (even if its real-part is not positive), and conclude that it is quite useful to check the diagonalizability of constraint propagation matrices. The discussion is general and can be applied to any numerical treatments of constrained dynamics.