TY - JOUR

T1 - Numerical relativity using a generalized harmonic decomposition

AU - Pretorius, Frans

N1 - Copyright:
Copyright 2005 Elsevier B.V., All rights reserved.

PY - 2005/1/21

Y1 - 2005/1/21

N2 - A new numerical scheme to solve the Einstein field equations based upon the generalized harmonic decomposition of the Ricci tensor is introduced. The source functions driving the wave equations that define generalized harmonic coordinates are treated as independent functions, and encode the coordinate freedom of solutions. Techniques are discussed to impose particular gauge conditions through a specification of the source functions. A 3D, free evolution, finite difference code implementing this system of equations with a scalar field matter source is described. The second-order-in-space-and-time partial differential equations are discretized directly without the use of first-order auxiliary terms, limiting the number of independent functions to 15 - ten metric quantities, four source functions and the scalar field. This also limits the number of constraint equations, which can only be enforced to within truncation error in a numerical free evolution, to four. The coordinate system is compactified to spatial infinity in order to impose physically motivated, constraint-preserving outer boundary conditions. A variant of the cartoon method for efficiently simulating axisymmetric spacetimes with a Cartesian code is described that does not use interpolation, and is easier to incorporate into existing adaptive mesh refinement packages. Preliminary test simulations of vacuum black-hole evolution and black-hole formation via scalar field collapse are described, suggesting that this method may be useful for studying many spacetimes of interest.

AB - A new numerical scheme to solve the Einstein field equations based upon the generalized harmonic decomposition of the Ricci tensor is introduced. The source functions driving the wave equations that define generalized harmonic coordinates are treated as independent functions, and encode the coordinate freedom of solutions. Techniques are discussed to impose particular gauge conditions through a specification of the source functions. A 3D, free evolution, finite difference code implementing this system of equations with a scalar field matter source is described. The second-order-in-space-and-time partial differential equations are discretized directly without the use of first-order auxiliary terms, limiting the number of independent functions to 15 - ten metric quantities, four source functions and the scalar field. This also limits the number of constraint equations, which can only be enforced to within truncation error in a numerical free evolution, to four. The coordinate system is compactified to spatial infinity in order to impose physically motivated, constraint-preserving outer boundary conditions. A variant of the cartoon method for efficiently simulating axisymmetric spacetimes with a Cartesian code is described that does not use interpolation, and is easier to incorporate into existing adaptive mesh refinement packages. Preliminary test simulations of vacuum black-hole evolution and black-hole formation via scalar field collapse are described, suggesting that this method may be useful for studying many spacetimes of interest.

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U2 - 10.1088/0264-9381/22/2/014

DO - 10.1088/0264-9381/22/2/014

M3 - Article

AN - SCOPUS:12944260516

VL - 22

SP - 425

EP - 451

JO - Classical and Quantum Gravity

JF - Classical and Quantum Gravity

SN - 0264-9381

IS - 2

ER -