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Linear wave testbed

This test checks the ability of a code to propagate the amplitude and phase of a traveling gravitational wave. The test is run in the linear regime where there are no complications due to the toroidal topology implicit in periodic boundary conditions. It reveals effects of numerical dissipation and other sources of inaccuracy in the evolution algorithm. We note that the evolution is meaningless once the accumulation of numerical error takes it out of the linear regime.

The initial 3-metric and extrinsic curvature $ K_{ij}$ are given by a diagonal perturbation with components

$\displaystyle ds^2= - dt^2 + dx^2 + (1+b) \, dy^2 + (1-b) \, dz^2,$ (1)

where

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

for a linearized plane wave traveling in the $ x$-direction. Here $ d$ is the linear size of the propagation domain, and the metric is written here in Gauss coordinates, i.e. with lapse $ \alpha=1$ and shift $ \beta^i=0$. The nontrivial components of extrinsic curvature are then

$\displaystyle K_{yy} = \frac{1}{2} \partial_t b, \quad K_{zz} = - \frac{1}{2} \partial_t b.$ (3)

As in the case of the gauge wave, by the simple coordinate transformation

$\displaystyle x = \frac{1}{\sqrt{2}}(y^\prime + x^\prime), \qquad y = \frac{1}{\sqrt{2}}(y^\prime - x^\prime)$ (4)

the propagation direction can be aligned with a diagonal. Setting $ d' = d \sqrt{2}$ to the size of the evolution domain in the $ x'$, $ y'$ directions gives periodicity along those directions.

The amplitude of the wave is chosen as $ A = 10^{-8}$, such that quadratic terms are of the order of numerical roundoff. Larger amplitudes mean that the solution does not stay in the linear regime sufficiently long.

The geometry of the grid is chosen identical to the 1D gauge wave test, with $ d=1$ in the 1D case and $ d'=\sqrt{2}$ in the diagonal case:

  • 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^i = -0.5 + (n-\frac{1}{2}) dx, \quad n=1\ldots 50\rho, \quad dx=1/(50\rho), \quad \rho \in \mathbb{Z}$
  • Time step: $ dt = dx/4 = 0.005 / \rho$

As in the gauge wave case, 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$, with output every crossing time. For the trivial directions ($ y$ and $ z$ for the wave propagating along the $ x$ axis and $ z$ for the wave propagating along the diagonal) we use the minimum number of grid points in the $ y,z$ directions that allow for non-trivial numerical second derivatives. For standard second order finite differencing this implies that we use 3 points in the appropriate directions. We run using $ \rho=1,2,4$.

Figure 1: A 1D linear wave shown at different resolutions. Although the run lasted 1000 crossing times, the output is shown after 500 crossing times in order to indicate the trend of how resolution effects phase accuracy. The numerical dissipation is low but the cumulative phase error is high at the coarser resolutions. It is clear that the phase error converges away. The results are from the code ABIGEL which implements a fully harmonic formulation [Szilagyi and Winicour(2002)].
\includegraphics[width=25pc]{LinearWave.eps}

The output quantities are similar to those for the gauge wave: the $ L_{\infty}$ and $ L_2$-norms, the maxima and minima, and profiles along the $ x$-axis through the center of the grid of $ g_{yy}$, $ g_{zz}$, $ tr(K)$, the Hamiltonian and any other nontrivial constraints, and the $ L_{\infty}$-norm of the difference from the linear exact solution for $ g_{zz}$. Figure 1 illustrates the profiles of $ g_{zz}-1$ obtained using a code based upon harmonic coordinates.

Bibliography

Szilagyi and Winicour(2002)
B. Szilagyi and J. Winicour (2002), gr-qc/0205044.

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Image taken from Universe Today
$Date: 2004/01/07 21:50:03 $