Numerical stability for finite difference approximations of Einstein’s equations

Abstract

We extend the notion of numerical stability of finite difference approximations to include hyperbolic systems that are first order in time and second order in space, such as those that appear in numerical relativity and, more generally, in Hamiltonian formulations of field theories. By analyzing the symbol of the second order system, we obtain necessary and sufficient conditions for stability in a discrete norm containing one-sided difference operators. We prove stability for certain toy models and the linearized Nagy-Ortiz-Reula formulation of Einstein's equations.We also find that, unlike in the fully first order case, standard discretizations of some well-posed problems lead to unstable schemes and that the Courant limits are not always simply related to the characteristic speeds of the continuum problem. Finally, we propose methods for testing stability for second order in space hyperbolic systems.

Iterated Crank-Nicolson method for hyperbolic and parabolic equations in numerical relativity

“…Most equations in numerical relativity can be recast in this form and more complex operators follow from these two cases (see [4] and references therein).…”

Stability of the puncture method with a generalized Baumgarte-Shapiro-Shibata-Nakamura formulation

“…III B we use characteristic variables to demonstrate well-posedness for the continuum system. The analysis is then extended to the semi-discrete case, and follows closely the method of [64,65]. Finally, we deal with the algebraic constraints in the fully-discrete system by demonstrating that the natural semi-discrete limit of the standard implementation (with constraint projection) is given by the systems considered in Sec III B.…”

“…For details we refer the reader to [64]. Discussion: A straightforward way to construct characteristic variables for the continuum system is to perform a 2 + 1 decomposition.…”

“…Following [64,66] we Fourier transform in space, and make a pseudodifferential reduction to first order. Spatial derivatives transform according to ∂ i → ıw i .…”

Extended black hole cosmologies in de Sitter space

“…In §3 we give some illustrative examples for binary black holes and binary cosmologies. In §4, we propose a hyperbolic system of equations for their evolution based on the 3+1 Hamiltonian equations of motion (Arnowitt, Deser & Misner 1962), where hyperbolicity generally facilitates stable numerical implementation (e.g., van Putten & Eardley (1996); Nagy, Ortiz, & Reula (2004) ;Calabrese, Hinder & Husa (2006) and references therein) by ensuring a real dispersion relation and hence stability whenever the Courant-Friedrichs-Lewy condition (Courant, Friedrichs & Lewy 1967) is satisfied. An outlook is included in §5.…”