A spectral solver for evolution problems with spatial -topology

Beyer, Florian

doi:10.1016/j.jcp.2009.05.037

Cited by 17 publications

(50 citation statements)

References 45 publications

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“…The bundle of orthonormal frames over S 2 and spin-weighted spherical harmonics SO(3) is the bundle of oriented orthonormal frames over S 2 with structure group U (1). Given that SO(3) is double covered by SU (2) and that the latter is diffeomorphic to S 3 , the Hopf map π : S 3 → S 2 can be identified with the bundle map. The theoretical details are discussed, for example, in [4].…”

Section: 3mentioning

confidence: 99%

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Numerical solutions of Einstein’s equations for cosmological spacetimes with spatial topologyS3and symmetry group U(1)

2016

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In this paper we consider the single patch pseudo-spectral scheme for tensorial and spinorial evolution problems on the 2-sphere presented in [3,4] which is based on the spin-weighted spherical harmonics transform. We apply and extend this method to Einstein's equations and certain classes of spherical cosmological spacetimes. More specifically, we use the hyperbolic reductions of Einstein's equations obtained in the generalized wave map gauge formalism combined with Geroch's symmetry reduction, and focus on cosmological spacetimes with spatial S 3 -topologies and symmetry groups U(1) or U(1) × U(1). We discuss analytical and numerical issues related to our implementation. We test our code by reproducing the exact inhomogeneous cosmological solutions of the vacuum Einstein field equations obtained in [7]. *

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Section: 3mentioning

confidence: 99%

“…As a consequence of this formalism, our code is (pseudo-)spectral in space; time evolutions are performed with the method of lines and standard ODE integrators (see below). We also point the reader to alternative implementations of this and similar formalisms in [2,10,25].…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions of Einstein’s equations for cosmological spacetimes with spatial topologyS3and symmetry group U(1)

2016

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“…Eventually, one can exploit the use of an appropriated basis functions adapted to the topology of the conformal time slices -see e.g. [8]-and reduce the evolution equations to a coupled system of ordinary differential equations in the time coordinate, which is then solved with spectral methods. The ultimate aim of this strategy is to depart from the linear equations and solve the full non-linear system of Einstein conformal field equations.…”

Section: Discussionmentioning

confidence: 99%

Spectral methods for the spin-2 equation near the cylinder at spatial infinity

Macedo

Kroon

2018

Class. Quantum Grav.

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We solve, numerically, the massless spin-2 equations, written in terms of a gauge based on the properties of conformal geodesics, in a neighbourhood of spatial infinity using spectral methods in both space and time. This strategy allows us to compute the solutions to these equations up to the critical sets where null infinity intersects with spatial infinity. Moreover, we use the convergence rates of the numerical solutions to read-off their regularity properties. *

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“…The use of spectral methods in relativistic evolutions can be traced back to pioneering work in the mid-1980s [66] (see also [67, 68, 213]). Over the last decade they have gained popularity, with applications in scenarios as diverse as relativistic hydrodynamics [313, 427, 428], characteristic evolutions [43], absorbing and/or constraint-preserving boundary conditions [314, 369, 365, 363], constraint projection [244], late time “tail” behavior of black-hole perturbations [382, 420], cosmological studies [19, 49, 50], extreme-mass-ratio inspirals within perturbation theory and self-forces [112, 162, 111, 425, 114, 113, 123] and, prominently, binary black-hole simulations (see, for example, [384, 329, 71, 381, 132, 288, 402, 131, 90, 289]) and black-hole-neutron-star ones [150, 168]. The method of lines (Section 7.3) is typically used with a small enough timestep so that the time integration error is smaller than the one due to the spatial approximation and spectral convergence is observed.…”

Section: Spatial Approximations: Spectral Methodsmentioning

confidence: 99%

Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations

Sarbach

Tiglio

2012

Living Rev. Relativ.

151

168

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Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black-hole problem within Einstein’s theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity.The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial boundary-value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein’s equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.

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A spectral solver for evolution problems with spatial -topology

Cited by 17 publications

References 45 publications

Numerical solutions of Einstein’s equations for cosmological spacetimes with spatial topologyS3and symmetry group U(1)

Numerical solutions of Einstein’s equations for cosmological spacetimes with spatial topologyS3and symmetry group U(1)

Spectral methods for the spin-2 equation near the cylinder at spatial infinity

Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations

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