Apples with Apples tests in Maya

Tyler Landis and Bernard Kelly, Penn State University

Introduction

The Apples with Apples project is the combined effort of several groups to establish common ground in Numerical Relativity. It seeks to answer basic questions about various aspects of numerical simulations -- questions like which formulation of the Einstein equations is most appropriate for astrophysically interesting simulations, and what are the best slicing and boundary conditions to use during these evolutions.

Penn State's Maya project is particpating in the Apples with Apples project. We have run tests on two types (sinewave and gaussian) of gauge waves in a periodic domain. Our evolution scheme is an implementation of the "BSSN" scheme.

The four-metric for both gauge waves was obtained from transforming the Minkowski metric, obtaining a four-metric of the form

four-metric-general

where H = H(x - t). This general form tells us that the lapse function, shift vector, and three-metric, respectively, are given by:
three-decomp-general



Also, as the shift vanishes, the extrinsic curvature can be calculated easily:

kxx-general

Our slicing choice was "harmonic" gauge, where the lapse function is evolved according to the Bona-Masso equation:

kxx-general

Sinusoidal Gauge Wave tests

In the sinusoidal case, the wave function was given by

h_sinusoid


We evolved this initial data -- with amplitude A = 0.01 and wavelength d = 1.0 -- on a periodic domain, where the long axis had x ∈ [ -0.5, 0.5 ]. Two resolutions are used: dx = 0.02 and dx = 0.01, implying 50 and 100 live grid points in the x direction, respectively, while only two grid points were used in the y- and z-directions, to give these negligible extent.

Below we show the metric function H(x - t) at four different times: t = 0, t = 25T, t = 40T, and t = 65T where T is the crossing time for the domain. In each case, the metric component is plotted for each of two different resolutions. As can be clearly seen, serious drift in the evolved quantities develops at an early time.
Figure 1: The metric function H for two different resolutions at times t = 0, t = 25T, t = 40T and t = 65T.

This drift is reflected in two-level convergence plots of the metric function error shown below. Convergence is slightly better than second order until the mid-twenties; after this point, we see a spike in the convergence factor. This is not a real improvement -- just a reflection of the breakdown of the coarser solution, which makes the still-accurate fine solution look better in comparison.

ADM codes evolving this same system have been run for 1000 T, though deviation from convergence is clear from early on. The BSSN system seems to do significantly worse, with runs blowing up before 100 T.
Figure 2: L2 norm of the error in the metric function H over coarse and fine resolutions.
Figure 3: L2 norm of the violation of the Hamiltonian constraint over time for coarse and fine resolutions.
Figure 4: 2-level convergence plot for Hamiltonian constraint L2 norms over time.

Gaussian Gauge Wave tests

In the gaussian case, the wave function was given by

h_gaussian



Again, we evolved this initial data on a periodic domain, with negligible y- and z-extents, but with x ∈ [ -1.0, 1.0 ], where the far-left and right points are identical by periodicity. The parameter a = 0.2. We should emphasize that this initial data is *not* periodic in nature, and evolution in a periodic domain may make little sense.

Below we show the metric function H(x - t) at five different times: t = 0, t = 3T, t = 6T, t = 15T, and t = 38T, where T is the crossing time for the domain. In each case, the metric component is plotted for each of three different resolutions. We notice that while the finer resolution keeps its shape, the coarser resolution run shows serious drift from the analytic solution after only a few crossing times.

Figure 5: The metric function H for three different resolutions at times t = 0, t = 3T, t = 6T, t = 15T and t = 38T.

Nevertheless, evolutions for this system lasted between 38 T and 45 T, depending on the resolution of the grid; the finer resolutions produced the longest runs.

A convergence plot was calculated for the (L2-norm of the) Hamiltonian constraint for the same three resolutions over the lifetime of the runs. As can be seen, convergence is at least second-order, well after the different resolution solutions lose all resemblance to each other.
Figure 6: L2 norm of the error in the metric function H over coarse, medium, and fine resolutions.
Figure 7: L2 norm of the violation of the Hamiltonian constraint over time for coarse, medium, and fine resolutions.
Figure 8: 3-level convergence plot for Hamiltonian constraint L2 norms over time.