Finite difference schemes for second order systems describing black holes

Motamed, Mohammad; Babiuc, Maria; Szilágyi, Béla; Kreiss, Heinz–Otto; Winicour, Jeffrey

doi:10.1103/physrevd.73.124008

Cited by 10 publications

(24 citation statements)

References 26 publications

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“…Such dissipative boundary conditions were first worked out in the one-dimensional (1D) case [40][41][42] and general results for the 3D case have been discussed recently in [23,43]. For a boundary with normal in the +x direction, such dissipative boundary conditions have the form…”

Section: Boundary Conditionsmentioning

confidence: 99%

“…In that case, when the energy E (t) is no longer a norm, stability can be based upon the positive energy associated with the time-like normal n µ to the Cauchy hypersurfaces, [43], where the global stability of a model black hole excision problem is treated. Although a stable boundary treatment for the superluminal algorithm (A.6) has been proposed [42], its extended stencil (due to the D Note that introduction of the auxiliary variableQ, which reduces the system to first-order in time, introduces no associated constraints.…”

Section: Appendix Blended Subluminal-superluminal Evolutionmentioning

confidence: 99%

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An explicit harmonic code for black-hole evolution using excision

Szilágyi

Pollney

Rezzolla

et al. 2007

Class. Quantum Grav.

108

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We describe an explicit in time, finite-difference code designed to simulate black holes by using the excision method. The code is based upon the harmonic formulation of the Einstein equations and incorporates several features regarding the well-posedness and numerical stability of the initialboundary problem for the quasilinear wave equation. After a discussion of the equations solved and of the techniques employed, we present a series of testbeds carried out to validate the code. Such tests range from the evolution of isolated black holes to the head-on collision of two black holes and then to a binary black hole inspiral and merger. Besides assessing the accuracy of the code, the inspiral and merger test has revealed that the marginally trapped surfaces contained within the common apparent horizon of the merged black hole can touch and even intersect. This novel feature in the dynamics of the marginally trapped surfaces is unexpected but consistent with theorems on the properties of these surfaces.

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Section: Boundary Conditionsmentioning

confidence: 99%

Section: Appendix Blended Subluminal-superluminal Evolutionmentioning

confidence: 99%

An explicit harmonic code for black-hole evolution using excision

Szilágyi

Pollney

Rezzolla

et al. 2007

Class. Quantum Grav.

108

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“…One can show 7 that in this case Re s 2 + γ −2 ω 2 Re(s ) which implies that Re(µ − ) < 0 < Re(µ + ). Therefore, the solution of (65) and (66) belonging to a trivial source term,f = 0, which decays as x → ∞ is given bỹ…”

Section: First-order Boundary Conditionsmentioning

confidence: 99%

“…Constraint-preserving boundary conditions for the harmonic system have been proposed before in [2][3][4][5] and tested numerically in [3,[6][7][8][9][10]. The boundary conditions of [2,3] are a combination of homogeneous Dirichlet and Neumann conditions, for which well posedness can be shown by standard techniques.…”

Section: Introductionmentioning

confidence: 99%

Outer boundary conditions for Einstein's field equations in harmonic coordinates

Ruiz¹,

Rinne²,

Sarbach³

2007

Class. Quantum Grav.

114

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We analyze Einstein's vacuum field equations in generalized harmonic coordinates on a compact spatial domain with boundaries. We specify a class of boundary conditions, which is constraint-preserving and sufficiently general to include recent proposals for reducing the amount of spurious reflections of gravitational radiation. In particular, our class comprises the boundary conditions recently proposed by Kreiss and Winicour, a geometric modification thereof, the freezing-0 boundary condition and the hierarchy of absorbing boundary conditions introduced by Buchman and Sarbach. Using the recent technique developed by Kreiss and Winicour based on an appropriate reduction to a pseudo-differential first-order system, we prove well posedness of the resulting initial-boundary value problem in the frozen coefficient approximation. In view of the theory of pseudo-differential operators, it is expected that the full nonlinear problem is also well posed. Furthermore, we implement some of our boundary conditions numerically and study their effectiveness in a test problem consisting of a perturbed Schwarzschild black hole.

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“…The approach which we introduce in this paper is partially derived from a method first discussed in a series of related papers by Kreiss, Winicour and collaborators [2,5,6], combined with the SBP energy method discussed in [7][8][9]. By deriving energy estimates for the semidiscrete system using the 'summation by parts' rule (defined below), one can ensure wellposedness [8,[10][11][12].…”

Section: Introductionmentioning

confidence: 99%