From geometry to numerics: interdisciplinary aspects in mathematical and numerical relativity

Jaramillo, José Luis; Kroon, Juan A. Valiente; Gourgoulhon, Éric

doi:10.1088/0264-9381/25/9/093001

Cited by 49 publications

(36 citation statements)

References 540 publications

(991 reference statements)

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“…Further details, including mathematical aspects of the equations and characteristic techniques, can be found in the review articles [113,92,114,115].…”

Section: Numerical Modeling Of Black Holesmentioning

confidence: 99%

Numerical simulations of black-hole binaries and gravitational wave emission

Sperhake

Berti

Cardoso

2013

Comptes Rendus. Physique

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We review recent progress in numerical relativity simulations of black-hole (BH) spacetimes. Following a brief summary of the methods employed in the modeling, we summarize the key results in three major areas of BH physics: (i) BHs as sources of gravitational waves (GWs), (ii) astrophysical systems involving BHs, and (iii) BHs in high-energy physics. We conclude with a list of the most urgent tasks for numerical relativity in these three areas.

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“…Further details, including mathematical aspects of the equations and characteristic techniques, can be found in the review articles [113,92,114,115].…”

Section: Numerical Modeling Of Black Holesmentioning

confidence: 99%

Numerical simulations of black-hole binaries and gravitational wave emission

Sperhake

Berti

Cardoso

2013

Comptes Rendus. Physique

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“…For example, the once perplexing simulations of colliding black holes are now routine applications that can be executed on moderate workstations. This is a result of a journey that spans decades of interdisciplinary work, drawing from seemingly divergent fields such as geometry, numerical analysis, software design, physics etc [55,50,29]. On the other hand, the field has reached a state of maturity that it can prove useful in other contexts such as gravity theories other than general relativity and in spacetime dimensions higher than four.…”

Section: Introductionmentioning

confidence: 99%

“…This is the essence of the formulation problem in numerical relativity [76,67]. In general, a typical recipe for a successful numerical evolution must take into account the following points: Initial data problem [33], Formulation problem [76], Boundary problem, Gauge problem and so on, See [50,5,13,41] for an in-depth discussion of these and other practical aspects of numerical relativity. It is conceivable that, when considering fourth order gravity, each of these areas will be affected in non-trivial ways by the accompanying new degrees of freedom.…”

Section: Introductionmentioning

confidence: 99%

On the hyperbolicity and stability of $$3+1$$ 3 + 1 formulations of metric f(R) gravity

Mongwane

2016

Gen Relativ Gravit

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3+1 formulations of the Einstein field equations have become an invaluable tool in Numerical relativity, having been used successfully in modeling spacetimes of black hole collisions, stellar collapse and other complex systems. It is plausible that similar considerations could prove fruitful for modified gravity theories. In this article, we pursue from a numerical relativistic viewpoint the 3 + 1 formulation of metric f (R) gravity as it arises from the fourth order equations of motion, without invoking the dynamical equivalence with Brans-Dicke theories. We present the resulting system of evolution and constraint equations for a generic function f (R), subject to the usual viability conditions. We confirm that the time propagation of the f (R) Hamiltonian and Momentum constraints take the same Mathematical form as in general relativity, irrespective of the f (R) model. We further recast the 3+1 system in a form akin to the BSSNOK formulation of numerical relativity. Without assuming any specific model, we show that the ADM version of f (R) is weakly hyperbolic and is plagued by similar zero speed modes as in the general relativity case. On the other hand the BSSNOK version is strongly hyperbolic and hence a promising formulation for numerical simulations in metric f (R) theories.

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“…Therefore, FCF is fundamentally different from fully hyperbolic formulations of GR and can be used as another check of the consistency of the numerical solutions of Einstein equations. There are other formulations which incorporate the constraints into the evolution system (see [56], section 5.2.2., for most relevant examples); however, no numerical simulations have been performed yet with most of them (e.g., [57]), or simulations are restricted to axisymmetric spacetimes (e.g., [58]). Moreover, FCF is a natural generalization of the CFC approximation; this fact makes possible a natural extension of all the numerical codes which use this approximation, in order to have a proper treatment of the gravitational radiation of the system without too much effort.…”

Section: Introductionmentioning

confidence: 99%

Gravitational waves in dynamical spacetimes with matter content in the fully constrained formulation

Cordero-Carrión

Cerdá–Durán²,

Ibáñez³

2012

Phys. Rev. D

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The Fully Constrained Formulation (FCF) of General Relativity (GR) is a framework introduced as an alternative to the hyperbolic formulations traditionally used in numerical relativity. The FCF equations form a hybrid elliptic-hyperbolic system of equations including explicitly the constraints. We present an implicit-explicit numerical algorithm to solve the hyperbolic part, whereas the elliptic sector shares the form and properties with the well known Conformally Flat Condition (CFC) approximation. We show the stability and convergence properties of the numerical scheme with numerical simulations of vacuum solutions. We have performed the first numerical evolutions of the coupled system of hydrodynamics and Einstein equations within FCF. As a proof of principle of the viability of the formalism, we present 2D axisymmetric simulations of an oscillating neutron star. In order to simplify the analysis we have neglected the back-reaction of the gravitational waves (GWs) into the dynamics, which is small (< 2%) for the system considered in this work. We use spherical coordinates grids which are well adapted for simulations of stars and allow for extended grids that marginally reach the wave zone. We have extracted the GW signature and compared to the Newtonian quadrupole and hexadecapole formulae. Both extraction methods show agreement within the numerical errors and the approximations used (∼ 30%).

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From geometry to numerics: interdisciplinary aspects in mathematical and numerical relativity

Cited by 49 publications

References 540 publications

Numerical simulations of black-hole binaries and gravitational wave emission

Numerical simulations of black-hole binaries and gravitational wave emission

On the hyperbolicity and stability of $$3+1$$ 3 + 1 formulations of metric f(R) gravity

Gravitational waves in dynamical spacetimes with matter content in the fully constrained formulation

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