Periodic Gauge Wave and Shifted Gauge Wave with Amp=0.5 Test Results for the ABIGEL Code

We present the results obtained with our harmonic evolution code ABIGEL, using two testbeds with periodic boundary conditions. The first is the gauge wave test, which is one of the core tests in the Apples with Apples suite of standardized tests. It is based upon transforming the Minkowski metric into

$$ds^2=-(1-H) dt^2 +(1-H)dx^2+dy^2+dz^2,$$ (1)

where

$$H = H(x-t)=A \sin \left( \frac{2 \pi (x - t)}{d} \right),$$ (2)

which describes a sinusoidal gauge wave of amplitude $A$ propagating along the $x$-axis. In order to test 2-dimensional features, we rotate the coordinates according to

$$x = \frac{1}{\sqrt{2}}(x^\prime - y^\prime), \qquad y = \frac{1}{\sqrt{2}}(x^\prime + y^\prime) \, .$$ (3)

which produces a gauge wave propagating along the diagonal with dependence

$$\sin \left( \frac{2 \pi (x' - y' - t' \sqrt{2})}{d'} \right), \quad \textrm{where} \quad d' = d \sqrt{2} \, .$$ (4)

Adjusting $d$ or $d'$ to the size of the evolution domain gives periodicity in the $x$ and $y$ directions.

The second testbed is a shifted version of the gauge wave, which tests the ability to deal with a non-zero shift as encountered in the black hole excision problem. The metric is obtained from the Minkowski metric $ds^2 =- d\hat t^2+d\hat x^2+d\hat y^2+d\hat z^2$ by the coordinate transformation

$$\begin{array}[c]{r c l} \hat t&=& t - \frac {Ad}{4\pi}\cos... ...}{d} \right), \\ \hat y&=& y, \\ \hat z&=& z \end{array}$$ (5)

where $d$ is the size of the evolution domain. This leads to the Kerr-Schild metric

$$ds^2=- dt^2 +dx^2+dy^2+dz^2 +H k_\alpha k_\beta dx^\alpha dx^\beta$$ (6)

where

$$k_\alpha=\partial_\alpha (x-t)$$ (7)

It describes a sinusoidal shifted gauge wave of amplitude $A$ propagating along the $x$-axis.

The simulation of the non-shifted gauge wave is complicated by excitation of the constraint preserving exponential modes.

The tests are run in both axis-aligned and diagonal form with amplitude $A=0.5$. We have found that smaller amplitudes are not as efficient for revealing problems. Larger amplitudes trigger gauge pathologies, e.g $g_{tt}\ge 0$, more quickly and may complicate code comparisons. The specified wave has wavelength $d=1$ in the axis-aligned simulation and wavelength $d'=\sqrt{2}$ in the diagonal simulation. We find that at least 50 grid points lead to resonable simulations for more than 10 crossing times and therefore make the following choices for the computational grid:

• Simulation domain:

  1D:	$\quad x \in [-0.5, +0.5],$	$\quad y = 0,$	$\quad z = 0,$	$\quad d=1$
  diagonal:	$\quad x \in [-0.5, +0.5],$	$\quad y \in [-0.5, +0.5],$	$\quad z = 0,$	$\quad d'=\sqrt{2}$

• Grid: $x_n = -0.5 + n dx, \quad n=0,1\ldots 50\rho, \quad dx=dy=dz=1/(50\rho), \quad \rho \in {Z}$
• Time step: $dt = dx/4 = 0.005 / \rho$

The 1D evolution is carried out for $T=1000$ crossing times, i.e.  $2\times10^5\rho$ time steps (or until the code crashes), with output every 10 crossing times. The 2D diagonal runs are carried out for $T=100$, with output every crossing time. We run using $\rho=1,2,4$.