Abstract
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.
Highlights
This review discusses fundamental tools from the analytical and numerical theory underlying the Einstein field equations as an evolution problem on a finite computational domain
Emphasis is placed on setting up the initial-boundary value problem (IBVP) for Einstein’s equations properly, by which we mean obtaining a well-posed formulation, which is flexible enough to incorporate coordinate conditions, which allow for long-term and accurate stable numerical evolutions
We generalize the theory one more step and consider evolution systems, which are described by quasilinear partial differential equations, that is, by nonlinear partial differential equations, which are linear in their highest-order derivatives
Summary
This review discusses fundamental tools from the analytical and numerical theory underlying the Einstein field equations as an evolution problem on a finite computational domain. Olivier Sarbach and Manuel Tiglio analyzing well-posedness of the IBVP For this reason, the theory of IBVP for hyperbolic problems is reviewed, followed by a presentation of the state of the art of boundary conditions for the harmonic and BSSN formulations of Einstein’s equations, where open problems related with gauge uniqueness are described. We refer the reader to [126] for a recent book on general relativity and the Einstein equations, which, among many other topics, discusses local and global aspects of the Cauchy problem, the constraint equations, and self-gravitating matter fields such as relativistic fluids and the relativistic kinetic theory of gases.
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