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A 3+1 Computational Scheme for Dynamic Spherically Symmetric Black Hole Spacetimes -- II: Time Evolution
General Relativity and Quantum Cosmology · Submitted 6 Jun 1999
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This is the second in a series of papers describing a 3+1 computational scheme for the numerical simulation of dynamic black hole spacetimes. | background 0.95 | 0.23 | 0.03 | asserted 0.99 |
| s1 | We discuss the numerical time-evolution of a given black-hole-containing initial data slice in spherical symmetry. | contribution 0.93 | 0.69 | 0.04 | asserted 0.78 |
| s2 | 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. | method 0.98 | 0.60 | 0.06 | asserted 1.00 |
| s3 | 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. | contribution 0.53 | 0.78 | 0.04 | asserted 0.78 |
| s4 | We use a free evolution, with Eddington-Finkelstein-like coordinates and the inner boundary placed at a fixed coordinate radius well inside the horizon. | method 1.00 | 0.89 | 0.04 | asserted 1.00 |
| s5 | 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. | method 1.00 | 0.88 | 0.04 | asserted 1.00 |
| s6 | These ODEs are then time-integrated by standard methods. | method 1.00 | 0.76 | 0.02 | asserted 0.99 |
| s7 | 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. | method 1.00 | 0.93 | 0.03 | asserted 1.00 |
| s8 | The spatial grid is smoothly nonuniform, but not adaptive. | method 0.91 | 0.88 | 0.03 | asserted 1.00 |
| s9 | 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. | result 0.99 | 0.97 | 0.70 | asserted 1.00 |
| s10 | 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. | result 1.00 | 0.96 | 0.27 | asserted 1.00 |
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Original 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.
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How the literature uses it
4 arXiv papers cite this in 7 sentences
Papers citing it · 2 sentences without a confident label
Uses its method or tool 1 paper
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On the other hand, at $r=r_ext$ we apply outgoing radiative boundary conditions [ref] .
Spherical non-linear absorption of cosmological scalar fields onto a black hole · Evolution · 0.97
Cites it as background 1 paper
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Excision, as suggested by Unruh to Thornburg [ref] and developed by many authors, see [ref] for a selection, relies on the idea that nothing can escape from the black hole region, so it should be possible to simply remove that region from the computational domain without affecting the domain of outer communication whatsoever.
An Implementation of DF-GHG with Application to Spherical Black Hole Excision · Introduction · 0.97
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