Implicit-explicit evolution of single black holes

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 time-stepping methods, for which the time-step 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) time scales characterizing the inspiral. Therefore, in explicit evolutions of binary black holes, the time-step size is typically orders of magnitude smaller than the relevant physical time scales. Implicit time-stepping methods allow for larger time steps, 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.

Figure 1

Error histories for GLoSISDC3. $\parallel ·\parallel$ represents the 1-norm with respect to the Cartesian coordinate measure over the spherical shell, i. e. $\parallel f\parallel ={\int }_V|f|dxdydz$, and $V$ is the improper (coordinate) volume of the spherical shell. As mentioned in the text, $\Delta {\psi }_{ab}$ denotes the difference between the numerical metric and the exact solution.

Figure 2

Error histories for GRrSISDC2. See the caption of Fig. 1 for an explanation of the figure labels.

Figure 3

Diagram for implicit sector of GLoSISDC3. The bottom plot depicts the cross section of the top plot along the imaginary axis, with $\lambda =\mathrm{i}\eta \in \mathrm{i}\mathbb{R}$.

Figure 4

Diagram for implicit sector of GRrSISDC2. See relevant comments given in the caption of Fig. 3.

Figure 5

Temporal convergence test. Error points (circles) have been computed using (28) in the text. The straight line in the plot and its indicated slope have been computed by a least-squares fit of the third through fifth error points.

Figure 6

Spatial convergence test. This plot depicts histories for the constraint energy norm $\sqrt{{\mathcal{E}}_c}$ described in the text.

Figure 7

Stability of various time-steppers when the constraint-damping terms are treated explicitly or implicitly. Plotted are constraint violations $\sqrt{{\mathcal{E}}_c}$. The top two panels show explicit treatment of the constraint-damping terms. This is stable for small time steps $\Delta t\le 1.024$ (top panel) and unstable for large time steps $\Delta t\ge 2.048$ (middle panel). The lowest panel shows implicit treatment of the constraint-damping term, resulting in stable evolutions for all time steps.

Figure 8

Demonstration of IMEX evolution of a single perturbed black hole using GRrSISDC2 with adaptive time-stepping. Top panel: the minimum and maximum of the horizon’s dimensionless intrinsic scalar curvature $M^{2}R$, which characterizes the horizon shape. Bottom panel: the Courant factor $\Delta t/\Delta x_{\min }$, where $\Delta t$ is the size of each time step and $\Delta x_{\min }$ is the minimum spacing between grid points, for an IMEX evolution and for an analogous explicit evolution. Both evolutions are evolved at the same spatial resolution (with approximately ${43}^3$ grid points).