127,555 research outputs found
Modeling the Black Hole Excision Problem
We analyze the excision strategy for simulating black holes. The problem is
modeled by the propagation of quasi-linear waves in a 1-dimensional spatial
region with timelike outer boundary, spacelike inner boundary and a horizon in
between. Proofs of well-posed evolution and boundary algorithms for a second
differential order treatment of the system are given for the separate pieces
underlying the finite difference problem. These are implemented in a numerical
code which gives accurate long term simulations of the quasi-linear excision
problem. Excitation of long wavelength exponential modes, which are latent in
the problem, are suppressed using conservation laws for the discretized system.
The techniques are designed to apply directly to recent codes for the Einstein
equations based upon the harmonic formulation.Comment: 21 pages, 14 postscript figures, minor contents updat
Constraint-preserving boundary treatment for a harmonic formulation of the Einstein equations
We present a set of well-posed constraint-preserving boundary conditions for
a first-order in time, second-order in space, harmonic formulation of the
Einstein equations. The boundary conditions are tested using robust stability,
linear and nonlinear waves, and are found to be both less reflective and
constraint preserving than standard Sommerfeld-type boundary conditions.Comment: 18 pages, 7 figures, accepted in CQ
The Well-posedness of the Null-Timelike Boundary Problem for Quasilinear Waves
The null-timelike initial-boundary value problem for a hyperbolic system of
equations consists of the evolution of data given on an initial characteristic
surface and on a timelike worldtube to produce a solution in the exterior of
the worldtube. We establish the well-posedness of this problem for the
evolution of a quasilinear scalar wave by means of energy estimates. The
treatment is given in characteristic coordinates and thus provides a guide for
developing stable finite difference algorithms. A new technique underlying the
approach has potential application to other characteristic initial-boundary
value problems.Comment: Version to appear in Class. Quantum Gra
Well-posed initial-boundary value problem for the harmonic Einstein equations using energy estimates
In recent work, we used pseudo-differential theory to establish conditions
that the initial-boundary value problem for second order systems of wave
equations be strongly well-posed in a generalized sense. The applications
included the harmonic version of the Einstein equations. Here we show that
these results can also be obtained via standard energy estimates, thus
establishing strong well-posedness of the harmonic Einstein problem in the
classical sense.Comment: More explanatory material and title, as will appear in the published
article in Classical and Quantum Gravit
Characteristic Evolution and Matching
I review the development of numerical evolution codes for general relativity
based upon the characteristic initial value problem. Progress in characteristic
evolution is traced from the early stage of 1D feasibility studies to 2D
axisymmetric codes that accurately simulate the oscillations and gravitational
collapse of relativistic stars and to current 3D codes that provide pieces of a
binary black hole spacetime. Cauchy codes have now been successful at
simulating all aspects of the binary black hole problem inside an artificially
constructed outer boundary. A prime application of characteristic evolution is
to extend such simulations to null infinity where the waveform from the binary
inspiral and merger can be unambiguously computed. This has now been
accomplished by Cauchy-characteristic extraction, where data for the
characteristic evolution is supplied by Cauchy data on an extraction worldtube
inside the artificial outer boundary. The ultimate application of
characteristic evolution is to eliminate the role of this outer boundary by
constructing a global solution via Cauchy-characteristic matching. Progress in
this direction is discussed.Comment: New version to appear in Living Reviews 2012. arXiv admin note:
updated version of arXiv:gr-qc/050809
Testing the well-posedness of characteristic evolution of scalar waves
Recent results have revealed a critical way in which lower order terms affect
the well-posedness of the characteristic initial value problem for the scalar
wave equation. The proper choice of such terms can make the Cauchy problem for
scalar waves well posed even on a background spacetime with closed lightlike
curves. These results provide new guidance for developing stable characteristic
evolution algorithms. In this regard, we present here the finite difference
version of these recent results and implement them in a stable evolution code.
We describe test results which validate the code and exhibit some of the
interesting features due to the lower order terms.Comment: 22 pages, 15 figures Submitted to CQ
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