Implicit-explicit (IMEX) evolution of single black holes

Abstract

Numerical simulations of binary black holes---an important predictive tool for the detection of gravitational waves---are computationally expensive, especially for binaries with high mass ratios or with rapidly spinning constituent holes. Existing codes for evolving binary black holes rely on explicit timestepping methods, for which the timestep size is limited by the smallest spatial scale through the Courant-Friedrichs-Lewy condition. Binary inspiral typically involves spatial scales (the spatial resolution required by a small or rapidly spinning hole) which are orders of magnitude smaller than the relevant (orbital, precession, and radiation-reaction) timescales characterizing the inspiral. Therefore, in explicit evolutions of binary black holes, the timestep size is typically orders of magnitude smaller than the relevant physical timescales. Implicit timestepping methods allow for larger timesteps, and they often reduce the total computational cost (without significant loss of accuracy) for problems dominated by spatial rather than temporal error, such as for binary-black-hole inspiral in corotating coordinates. However, fully implicit methods can be difficult to implement for nonlinear evolution systems like the Einstein equations. Therefore, in this paper we explore implicit-explicit (IMEX) methods and use them for the first time to evolve black-hole spacetimes. Specifically, as a first step toward IMEX evolution of a full binary-black-hole spacetime, we develop an IMEX algorithm for the generalized harmonic formulation of the Einstein equations and use this algorithm to evolve stationary and perturbed single-black-hole spacetimes. Numerical experiments explore the stability and computational efficiency of our method.

14 pages, 8 figures

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• Black-hole binaries, gravitational waves, and numerical relativity
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• LISA technology and instrumentation
• Orbital Evolution of Extreme-Mass-Ratio Black-Hole Binaries with Numerical Relativity
• Testing outer boundary treatments for the Einstein equations
• Simulating merging binary black holes with nearly extremal spins
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• Suitability of post-Newtonian/numerical-relativity hybrid waveforms for gravitational wave detectors
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• The Science of the Einstein Telescope
• Numerical Relativity of Compact Binaries in the 21st Century
• Exploring the Outer Limits of Numerical Relativity
• A scalable elliptic solver with task-based parallelism for the SpECTRE numerical relativity code
• A hp-adaptive discontinuous Galerkin solver for elliptic equations in numerical relativity
• Quasistationary hair for binary black hole initial data in scalar Gauss-Bonnet gravity