A 3+1 Computational Scheme for Dynamic Spherically Symmetric Black Hole Spacetimes -- II: Time Evolution
arXiv:gr-qc/9906022
Abstract
This is the second in a series of papers describing a 3+1 computational scheme for the numerical simulation of dynamic black hole spacetimes. We discuss the numerical time-evolution of a given black-hole-containing initial data slice in spherical symmetry. We avoid singularities via the "black-hole exclusion" or "horizon boundary condition" technique, where the slices meet the black hole's singularity, but on each slice a spatial neighbourhood of the singularity is excluded from the domain of the numerical computations. After first discussing some of the key design choices which arise with the black hole exclusion technique, we then give a detailed description of our numerical evolution scheme for spherically symmetric scalar field evolution, assuming that a black hole is already present on the initial slice. We use a free evolution, with Eddington-Finkelstein-like coordinates and the inner boundary placed at a fixed coordinate radius well inside the horizon. Our numerical scheme is based on the method of lines (MOL), where spacetime PDEs are first finite differenced in space only, yielding a system of coupled ODEs for the time evolution of the field variables along the spatial-grid-point world lines. These ODEs are then time-integrated by standard methods. We use 4th order finite differencing in both space and time, with 5 and/or 6 point spatial molecules (off-centered near the grid boundaries), and a Runge-Kutta time integrator. The spatial grid is smoothly nonuniform, but not adaptive. We present numerical black hole + scalar field evolutions showing that this scheme is stable, can evolve "forever" (we have gone to t > 4000m), and is very accurate. At a resolution Delta_r/r = 3% near the horizon, typical errors in g_ij(K_ij) at t=100m are <= 1e-5(3e-7), and the energy constraint is < 3e-5.
63 pages, 22 pages postscript figures + 1 postscript rotated table, uses REVTeX and natbib, revised to properly normalize constraints, discuss variant finite differencing scheme, and discuss problems with noise addition; now submitted to Physical Review D
References in corpus (21)
- Late-Time Behavior of Stellar Collapse and Explosions: I. Linearized Perturbations
- Gravitational Energy in Spherical Symmetry
- Quasi-Local Gravitational Energy
- Late-Time Behavior of Stellar Collapse and Explosions: II. Nonlinear Evolution
- Collapse to Black Holes in Brans-Dicke Theory: II. Comparison with General Relativity
- Towards a Singularity-Proof Scheme in Numerical Relativity
- Finding Apparent Horizons in Numerical Relativity
- Collapse to Black Holes in Brans-Dicke Theory: I. Horizon Boundary Conditions for Dynamical Spacetimes
- Boosted three-dimensional black-hole evolutions with singularity excision
- Black Hole--Scalar Field Interactions in Spherical Symmetry
- Dynamics of Apparent and Event Horizons
- Implementing an apparent-horizon finder in three dimensions
- Adaptive mesh and geodesically sliced Schwarzschild spacetime in 3+1 dimensions
- Finding Apparent Horizons in Dynamic 3D Numerical Spacetimes
- Event Horizons in Numerical Relativity I: Methods and Tests
- Numerical Evolution of Black Holes with a Hyperbolic Formulation of General Relativity
- Calculation of gravitational wave forms from black hole collisions and disk collapse: Applying perturbation theory to numerical spacetimes
- Cauchy-characteristic Evolution of Einstein-Klein-Gordon Systems
- The Constraints in Spherically Symmetric General Relativity I --- Optical Scalars, Foliations, Bounds on the Configuration Space Variables and the Positivity of the Quasi-Local Mass
- A 3+1 Computational Scheme for Dynamic Spherically Symmetric Black Hole Spacetimes -- I: Initial Data
- Adaptative Mesh Refinement in Numerical Relativity