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Spectral Methods for Numerical Relativity
 LIV. REV. RELAT., SUBMITTED
, 2007
"... Equations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods where, typically, the vari ..."
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Equations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods where, typically, the various functions are expanded onto sets of orthogonal polynomials or functions. A theoretical introduction on spectral expansion is first given and a particular emphasis is put on the fast convergence of the spectral approximation. We present then different approaches to solve partial differential equations, first limiting ourselves to the onedimensional case, with one or several domains. Generalization to more dimensions is then discussed. In particular, the case of time evolutions is carefully studied and the stability of such evolutions investigated. One then turns to results obtained by various groups in the field of General Relativity by means of spectral methods. First, works which do not involve explicit timeevolutions are discussed, going from rapidly rotating strange stars to the computation of binary black holes initial data. Finally, the evolutions of various systems of astrophysical interest are presented, from supernovae core collapse to binary black hole mergers.
Discrete differential forms for cosmological spacetimes
 SIAM J.Sci.Comput
, 2010
"... Abstract. In this article we describe applications of the numerical method of discrete differential forms in computational GR. In particular we consider the initial value problem for vacuum spacetimes that admit plane gravitational waves. As described in an earlier paper the discrete differential f ..."
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Abstract. In this article we describe applications of the numerical method of discrete differential forms in computational GR. In particular we consider the initial value problem for vacuum spacetimes that admit plane gravitational waves. As described in an earlier paper the discrete differential form approach can provide accurate results in spherically symmetric spacetimes [28]. Moreover it is manifestly coordinate independent. Here we use the polarised Gowdy solution as a testbed for two numerical schemes. One scheme reproduces that solution very well, in particular it is stable for a comparatively long time and converges quadratically. 1.
Exploring New Physics Frontiers Through Numerical Relativity
"... The demand to obtain answers to highly complex problems within strongfield gravity has been met with significant progress in the numerical solution of Einstein’s equations – along with some spectacular results – in various setups. We review techniques for solving Einstein’s equations in generic spa ..."
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The demand to obtain answers to highly complex problems within strongfield gravity has been met with significant progress in the numerical solution of Einstein’s equations – along with some spectacular results – in various setups. We review techniques for solving Einstein’s equations in generic spacetimes, focusing on fully nonlinear evolutions but also on how to benchmark those results with perturbative approaches. The results address problems in highenergy physics, holography, mathematical physics, fundamental physics, astrophysics and cosmology. 1 ar X iv:1
Numerical treatment of interfaces for secondorder wave equations
, 2011
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