Novel finite-differencing techniques for numerical relativity: application to black-hole excision

Abstract

We use rigorous techniques from numerical analysis of hyperbolic equations in bounded domains to construct stable finite-difference schemes for Numerical Relativity, in particular for their use in black hole excision. As an application, we present 3D simulations of a scalar field propagating in a Schwarzschild black hole background.The numerical implementation of Einstein's equations represents a daunting task. The involved nature of the equations themselves, and a number of technical difficulties (related to the necessarily finite computational domain, the limitations in relative computational power and the need to deal with singularities, constraints and gauge freedom) imply a significant challenge.Numerical solutions of Einstein's equations involve solving a nonlinear set of partial differential equations on a bounded domain, and formally constitute an initialboundary-value problem (IBVP). Constructing stable and long term well behaved numerical approximations for such systems with boundaries is highly non-trivial. Here the term numerical stability is used in the sense that is equivalent, through Lax's theorem, to convergence (that is, the property that the numerical solution will approach the continuum one when resolution is increased). Delicate complications arise due to corner and edges at outer (and possible inner) boundaries, all of which introduce subtleties for a stable implementation.Recently, however, several sophisticated tools of rigorous numerical analysis have been developed for systematically constructing stable numerical schemes for IBVP's. They employ a discrete form of well-posedness and thus are stable by construction, at least for linear systems. At this time, these tools have essentially not been used by the numerical relativity community. The purpose of this paper is to outline their use, in particular for black hole excision.An IBVP consists of three ingredients: a partial differential equation, initial and boundary data. It is wellposed if a solution exists, is unique, and depends continuously on the initial and boundary data. It is well known that stable finite difference schemes approximating an IBVP can only be constructed for well-posed systems. While problems in general relativity are typically nonlinear we consider here the linear IBVP, as stability in the linearized case is a necessary condition for stability in the full nonlinear system and these methods may also be applied to nonlinear equations. Consider the linear IBVP on a domain [0, ∞) × Ωwhere u is a vector-valued function, w + and w − are incoming and outgoing modes, and S is sufficiently small (maximally dissipative boundary conditions [3]). The system is assumed to be symmetric hyperbolic, i.e., there exists a symmetric positive definite matrix H(t, x), the symmetrizer, satisfying HA i = (HA i ) T . The usual proof of well-posedness proceeds by defining an "energy", E = Ω u T Hu d 3 x , and by showing that E can be bounded as a function of the initial and boundary data. Analogously, a way to construct stable numerical s...

AMR, stability and higher accuracy

“…A clean way to approach this problem is to consider operators satisfying summation by parts (SBP) which have recently been incorporated successfully in strongly/symmetric hyperbolic systems in numerical relativity [3,4,10,11,12]. These techniques guarantee that a large set of problems can be implemented stably.…”

3D simulations of Einstein's equations: Symmetric hyperbolicity, live gauges, and dynamic control of the constraints

“…That is, by constructing a numerically stable scheme for the IBVP under consideration. One way of doing so is by constructing difference operators and imposing the discrete boundary conditions in a way such that the steps followed at the continuum to show well posedness can be reproduced at the discrete level [4,5,6].…”

“…Section IV briefly summarizes the numerical techniques used in this paper, already presented in Ref. [6], and the details of the test-beds here studied. One of these test-beds is the study of a periodic gauge wave, presented in Section V. There we show that the use of a symmetric hyperbolic formulation and a small amount of artificial dissipation suffices to evolve this solution with rather large amplitudes and for long times.…”

Recent Analytical and Numerical Techniques Applied to the Einstein Equations

“…Additionally, when dealing with non-trivial domains containing inner boundaries, additional complexities must be addressed to attain SBP, see Ref. [4]. The finite operator D is then used for the discretization of the spatial derivatives in the evolution equations, thus obtaining a semi-discrete system.…”

“…Explicit expressions for such dissipation operators are presented in Ref. [4]. To summarize, beginning with a well-posed initial-boundary value problem, we mimic the derivation of continuum energy estimates for the discrete problem using (1) spatial derivative operators satisfying summation by parts, (2) orthogonal projections to represent boundary conditions and (3) choosing an appropriate time integrator.…”