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

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.

23 pages, RevTeX4, many figures. Revised version. Added subtitle, reduced figures, rephrased introduction, and a native checked. :-)

References in corpus (5)

• Numerical Relativity: A review
• Extending the lifetime of 3D black hole computations with a new hyperbolic system of evolution equations
• A cure for unstable numerical evolutions of single black holes: adjusting the standard ADM equations
• Constraint propagation in the family of ADM systems
• On the Effect of Constraint Enforcement on the Quality of Numerical Solutions in General Relativity

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• Constraint Propagation of $C^2$-adjusted Formulation II --- Another Recipe for Robust Baumgarte-Shapiro-Shibata-Nakamura Evolution System ---
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