Adjusted ADM systems and their expected stability properties: constraint propagation analysis in Schwarzschild spacetime

TL;DR

This paper analyzes the stability of adjusted ADM systems in Schwarzschild spacetime by studying constraint propagation, aiming to improve long-term numerical simulations of Einstein's equations.

Contribution

It provides a detailed analysis of constraint propagation in Schwarzschild spacetime and explores how adjustments affect stability and constraint violations.

Findings

Eigenvalues depend on coordinate choices and adjustments.

Detweiler's proposal is effective but has issues near the horizon.

Analysis guides future improvements in numerical relativity simulations.

Abstract

In order to find a way to have a better formulation for numerical evolution of the Einstein equations, we study the propagation equations of the constraints based on the Arnowitt-Deser-Misner formulation. By adjusting constraint terms in the evolution equations, we try to construct an "asymptotically constrained system" which is expected to be robust against violation of the constraints, and to enable a long-term stable and accurate numerical simulation. We first provide useful expressions for analyzing constraint propagation in a general spacetime, then apply it to Schwarzschild spacetime. We search when and where the negative real or non-zero imaginary eigenvalues of the homogenized constraint propagation matrix appear, and how they depend on the choice of coordinate system and adjustments. Our analysis includes the proposal of Detweiler (1987), which is still the best one according to our conjecture but has a growing mode of error near the horizon. Some examples are snapshots of a maximally sliced Schwarzschild black hole. The predictions here may help the community to make further improvements.