Improved gauge driver for the generalized harmonic Einstein system

A new gauge driver is introduced for the generalized harmonic (GH) representation of Einstein’s equation. This new driver allows a rather general class of gauge conditions to be implemented in a way that maintains the hyperbolicity of the combined evolution system. This driver is more stable and effective and, unlike previous drivers, allows stable evolutions using the dual-frame evolution technique. Appropriate boundary conditions for this new gauge driver are constructed, and a new boundary condition for the “gauge” components of the spacetime metric in the GH Einstein system is introduced. The stability and effectiveness of this new gauge driver are demonstrated through numerical tests, which impose a new damped-wave gauge condition on the evolutions of single black-hole spacetimes.

Figure 1

Coordinate radius of the apparent horizon of the black hole $R_H$ as it evolves under the effects of the dynamically driven gauge. This test uses a single-frame evolution with gauge-driver parameters $\mu =\eta =16$.

Figure 2

Constraints of the GH Einstein system $\Vert {\mathcal{C}}_{\mathrm{GH}}\Vert$ for a single-frame evolution of a Schwarzschild black hole with strongly perturbed lapse and shift. This test uses a single-frame evolution with gauge-driver parameters $\mu =\eta =16$, and several different values of the numerical resolution parameters $N_r$ and $L$. The small inset graph contains a magnified view of $\Vert {\mathcal{C}}_{\mathrm{GH}}\Vert$ during the time interval $15M\le t\le 30M$, showing that the solution is convergent during this most dynamical part of the evolution.

Figure 3

Curves show $|\delta M|/M$, the deviations in the mass of the hole from its initial value. This test uses a single-frame evolution with gauge-driver parameters $\mu =\eta =16$.

Figure 4

Effectiveness of the gauge-driver equation is demonstrated by showing $\Vert H-F\Vert /\Vert F\Vert$ for an evolution of a Schwarzschild black hole with strongly perturbed lapse and shift. This test uses a single-frame evolution with gauge-driver parameters $\mu =\eta =16$.

Figure 5

Effectiveness of the gauge-driver system is demonstrated for various values of the gauge-driver parameters $\eta$ and $\mu$. This test uses a dual-frame evolution method with the comoving frame rotating with respect to the inertial frame at angular velocity $\Omega =1/M$.