Energy Norms and the Stability of the Einstein Evolution Equations
arXiv:gr-qc/0206035 · doi:10.1103/PhysRevD.66.084014
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.
Corrected typos; to appear in Physical Review D
References in corpus (11)
- Extending the lifetime of 3D black hole computations with a new hyperbolic system of evolution equations
- Black Hole Excision for Dynamic Black Holes
- Well-Posed Initial-Boundary Evolution in General Relativity
- Constraint-preserving boundary conditions in numerical relativity
- Advantages of modified ADM formulation: constraint propagation analysis of Baumgarte-Shapiro-Shibata-Nakamura system
- Exploiting gauge and constraint freedom in hyperbolic formulations of Einstein's equations
- Illustrating Stability Properties of Numerical Relativity in Electrodynamics
- Numerical stability of a new conformal-traceless 3+1 formulation of the Einstein equation
- Adjusted ADM systems and their expected stability properties: constraint propagation analysis in Schwarzschild spacetime
- Stability properties of a formulation of Einstein's equations
- First-order symmetrizable hyperbolic formulations of Einstein's equations including lapse and shift as dynamical fields
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- Controlling the Growth of Constraints in Hyperbolic Evolution Systems
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- A hyperbolic tetrad formulation of the Einstein equations for numerical relativity
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