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Robust stability testbed

The robust stability testbed [Szilagyi et al.(2000)Szilagyi, Gómez, Bishop, and Winicour] efficiently reveals exponentially growing modes which otherwise might be masked beneath a strong initial signal for a considerable evolution time. It is based upon small random perturbations of Minkowski space. The initial data consists of random numbers $ \epsilon$ applied as a perturbation at each grid point to every code variable requiring initialization. For example, the initial 3-metric is initialized as $ h_{ij}=\delta_{ij}+\epsilon_{ij}$, where the $ \epsilon_{ij}$ are independent random numbers. The range of the random numbers ensures that $ \epsilon^2$ effects are below roundoff accuracy so that the evolution remains in the linear domain unless instabilities arise.

For economy, we fix the following parameters:

  • Simulation domain: $ x \in [-0.5, +0.5]$
  • Grid: $ x_n = -0.5 + (n-\frac{1}{2}) dx, \quad n=1\ldots 50\rho, \quad dx=dy=dz=1/(50\rho), \quad \rho \in \mathbb{Z}$
  • Time step: $ dt = dx/2 = 0.01 / \rho$
The parameter $ \rho$ allows for refinement testing, which clarifies the results as can be seen in Figure 1. The values used in that test were $ \rho=1,2,4$. We use the minimum number of grid points in the $ y,z$ directions that allow for non-trivial numerical second derivatives - this means we carry out this test in a long channel rather than a cube. For standard second order finite differencing this implies that we use 3 points in the $ y$ and $ z$ directions.

The amplitude of the random noise should be scaled with the grid spacing as

$\displaystyle \epsilon \in (-10^{-10}/\rho^2,+10^{-10}/\rho^2).$ (1)

This ensures that the norm of the Hamiltonian constraint violation in the initial data will be (on average) the same for different values of the refinement factor $ \rho$. This means that in the continuum limit we will have a solution that is not a solution of the Einstein equations but ``close'' to one. This would be the case in a real numerical evolution where machine precision takes the place of $ \epsilon$. If a code cannot stably evolve the random noise then it will be unable to evolve a real initial data set.

The test should be run for a time of $ T=1000$ (corresponding to $ 1000$ crossing times) or until the code crashes. Performance is monitored by outputting the $ L_\infty$ norm of the Hamiltonian constraint once per crossing time, i.e. at $ t=0,1,2,3,...$. Because the initial data violates the constraints, any instability can be expected to lead to an exponential growth of the Hamiltonian unless enforcement of the Hamiltonian constraint were built into the evolution algorithm. As an example, Figure 1 shows the performance of standard ADM and BSSN codes and illustrates the efficacy of this test in revealing unstable codes.

Figure 1: Left: the robust stability test applied to the standard formulation of the ADM equations. It is clear that there is an exponentially growing mode and that the growth rate of the mode depends on resolution. Right: the robust stability test applied to the BSSN formulation. The violation of the constraint is approximately constant even after 1000 crossing times. In both cases the harmonic gauge was used. Note the differences in the axes.
\includegraphics[width=40pc]{RobustStability2.eps}

Bibliography

Szilagyi et al.(2000)Szilagyi, Gómez, Bishop, and Winicour
B. Szilagyi, R. Gómez, N. T. Bishop, and J. Winicour, Phys. Rev. D 62 (2000), gr-qc/9912030.

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$Date: 2004/01/07 21:50:03 $