Advantages of modified ADM formulation: constraint propagation analysis of Baumgarte-Shapiro-Shibata-Nakamura system

Abstract

Several numerical relativity groups are using a modified ADM formulation for their simulations, which was developed by Nakamura et al (and widely cited as Baumgarte-Shapiro-Shibata-Nakamura system). This so-called BSSN formulation is shown to be more stable than the standard ADM formulation in many cases, and there have been many attempts to explain why this re-formulation has such an advantage. We try to explain the background mechanism of the BSSN equations by using eigenvalue analysis of constraint propagation equations. This analysis has been applied and has succeeded in explaining other systems in our series of works. We derive the full set of the constraint propagation equations, and study it in the flat background space-time. We carefully examine how the replacements and adjustments in the equations change the propagation structure of the constraints, i.e. whether violation of constraints (if it exists) will decay or propagate away. We conclude that the better stability of the BSSN system is obtained by their adjustments in the equations, and that the combination of the adjustments is in a good balance, i.e. a lack of their adjustments might fail to obtain the present stability. We further propose other adjustments to the equations, which may offer more stable features than the current BSSN equations.

10 pages, RevTeX4, added related discussion to gr-qc/0209106, the version to appear in Phys. Rev. D

References in corpus (7)

• Numerical Relativity: A review
• Extending the lifetime of 3D black hole computations with a new hyperbolic system of evolution equations
• Gravitational Waves from the Merger of Binary Neutron Stars in a Fully General Relativistic Simulation
• Improved numerical stability of stationary black hole evolution calculations
• 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
• On the Effect of Constraint Enforcement on the Quality of Numerical Solutions in General Relativity

Cited by in corpus (28)

• Three-dimensional relativistic simulations of rotating neutron-star collapse to a Kerr black hole
• Numerical Relativity and Compact Binaries
• Binary black-hole evolutions of excision and puncture data
• General-covariant evolution formalism for Numerical Relativity
• Toward standard testbeds for numerical relativity
• Relativistic Hydrodynamic Evolutions with Black Hole Excision
• Black hole head-on collisions and gravitational waves with fixed mesh-refinement and dynamic singularity excision
• Energy Norms and the Stability of the Einstein Evolution Equations
• Towards a gauge-polyvalent Numerical Relativity code
• Black hole critical behaviour with the generalized BSSN formulation
• Constraint Propagation of $C^2$-adjusted Formulation - Another Recipe for Robust ADM Evolution System
• Algebraic stability analysis of constraint propagation
• Improving the convergence order of binary neutron star merger simulations in the Baumgarte-Shapiro-Shibata-Nakamura formulation
• Improved Moving-Puncture Techniques for Compact Binary Simulations
• Non-linear metric perturbation enhancement of primordial gravitational waves
• A spectral method algorithm for numerical simulations of gravitational fields
• On the Hyperbolicity and Stability of $3+1$ Formulations of Metric $f(R)$ Gravity
• Constraint propagation in N+1-dimensional space-time
• Improved fast-rotating black hole evolution simulations with modified Baumgarte-Shapiro-Shibata-Nakamura formulation
• Numerical experiments of adjusted BSSN systems for controlling constraint violations
• Constraint Propagation of $C^2$-adjusted Formulation II --- Another Recipe for Robust Baumgarte-Shapiro-Shibata-Nakamura Evolution System ---
• Gravitational Instability of Yang-Mills Cosmologies
• Modifications for numerical stability of black hole evolution
• On constraint preservation and strong hyperbolicity
• New frontiers in Numerical Relativity
• On the stability of covariant BSSN formulation
• Geometrical optics analysis of the short-time stability properties of the Einstein evolution equations
• Stable numerical simulation of Einstein equations in gravitational collapse space--time