Evolutions of Gowdy, Brill, and Teukolsky initial data on a smooth lattice

Numerical results, based on a lattice method for computational general relativity, will be presented for Cauchy evolution of initial data for the Brill, Teukolsky and polarized Gowdy spacetimes. The simple objective of this paper is to demonstrate that the lattice method can, at least for these spacetimes, match results obtained from contemporary methods. Some of the issues addressed in this paper include the handling of axisymmetric instabilities (in the Brill space-time) and an implementation of a Sommerfeld radiation condition for the Brill and Teukolsky spacetimes. It will be shown that the lattice method performs particularly well in regard to the passage of the waves through the outer boundary. Questions concerning multiple black holes, mesh refinement and long term stability will not be discussed here but may form the basis of future work.

Figure 1

Two examples of a subset of the Gowdy one-dimensional lattice. The left figure shows a single cell in the while the right figure shows a pair of neighboring cells. The purple vertices are the central vertices of their respective cells. Note that the vertical legs pass through the central vertex and begin and end on the red vertices. This also applies to the corresponding horizontal legs. In contrast, the radial legs begin and end on the central vertices.

Figure 2

Details of the Brill two-dimensional lattice. The left figure shows a subset of the lattice including two overlapping cells. Each cell is a $2\times 2$ set of vertices and legs. An axisymmetric lattice is obtained by assembling copies of the two-dimensional lattice in the manner shown in the middle figure. The yellow legs in the middle figure are needed to define the separation between the copies. The right figure shows the various subsets of the lattice used to evolve the data and to apply various boundary conditions. Data in the outer boundary (the orange region) were evolved using a radiation boundary condition while the data on and near the symmetry axis (the dark blue region) were evolved by interpolating the time derivatives from the nearby cells (the light blue region). The remaining data (in the yellow region) were evolved using the lattice evolution equations.

Figure 3

A typical set of vertices and legs used in computing the transition matrices, ${m^a}_{bc}$. The coordinate axes in these figures are applicable only to the two-dimensional Brill lattice and should be ignored when reading the discussion in Appendix pp1 particularly in the calculations leading to equation (a16).

Figure 4

A typical computational cell for the Teukolsky lattice. This figure shows, for simplicity, only one of three sets of yellow diagonal legs. A proper figure would show yellow diagonal legs on each of the three coordinate planes (bounded by the green rectangles). Note also that though this cell looks regular (roughly equal leg lengths and apparently orthogonal legs) this is again just to simplify the figure. In general the leg lengths and their mutual angles will vary (slightly) across the cell.

Figure 5

This figure shows the rapid expansion (into the future of the $t=0$ singularity) of the lattice in the $1+\log$ slicing. The left plot shows the lapse (from $t=1$ to $t=20$ in steps of 1) as a function of the unscaled proper distance while the right plot shows the same data but using a rescaled $z$-axis. The red curves display the lattice data (for $N_z=1024$) while the blue dots are from the Cactus data (with $N_z=400$ though only every fourth point is shown). The agreement between the lattice and Cactus data is very good.

Figure 6

A comparison of the lattice data for the exact slicing against the New-Watt et al. [30] data. The continuous line denotes the lattice data (using $N_z=1024$) while the New-Watt data (with $N_z=32$) are denoted by points. It is clear that the lattice data agrees very well with the New-Watt data. There are nine curves in each figure representing data from $t=2$ to $t=6$ in steps of 0.5.

Figure 7

This figure is similar to the previous figure but this time for the $1+\log$ slicing. The Cactus data (blue points) is based on $N_z=400$ with only every fourth point shown. The lattice data (red lines) is based on $N_z=1024$. Each figure contains 20 curves for $t=2$ to $t=20$ in steps of 1.

Figure 8

This figure shows the behavior, in the $1+\log$ slicing, of the $C_1$ constraint (71) over time (left panel) and across the grid at a fixed time (right panel). The data in the left panel are for the case $N_z=1024$ and show the maximum values of $C_1$ across the grid. The right-hand panel shows three curves, $N_z=256$ (red), $N_z=512$ (blue) and $N_z=1024$ (green) with $y$ values, at $t=5$, scaled by 1, 32 and 1024 respectively. The close agreement in the curves suggests that the constraints converge to zero as $\mathcal{O}(N_z^{-5})$. The fifth order convergence is unexpected and is likely due to a fortuitous interplay between the fourth and fifth order terms in the truncation errors (arising from the fourth order Runge-Kutta integrations and the fifth order Kreiss-Oliger dissipation). Similar behavior was observed for the remaining two constraints (72), (73). The somewhat erratic behavior in the left panel most likely arises by the fact that the grid point on which the maximum occurs need not be a continuous function of time.

Figure 9

This figure show the convergence of two metric functions, $K_{zz}$ and $R_{xyxy}$, as a function of $N_z$ in the $1+\log$ slicing. The three curves correspond to $N_z=128$ (red), $N_z=256$ (blue) and $N_z=512$ (green) and have their $y$ values scaled by 1, 32 and 1024 respectively. For the $1+\log$ slicing there is no exact solution available so the best available data (i.e., $N_z=1024$) was taken as a best estimate of the exact solution. This suggests that the lattice data is converging to the exact solution as $\mathcal{O}(N_z^{-5})$. As with the previous figure, the fifth order convergence is unexpected and is likely due to the same interplay between the truncation error terms as noted before.

Figure 10

This figure shows a comparison between the lattice, ADM and BSSN evolutions of $R_{xyxy}$ for the Brill initial data at $t=5$. All three methods agree well though the ADM and BSSN results show small waves near the symmetry axis. The figure in the lower right shows the data for all three methods (red, lattice), (blue, ADM) and (green, BSSN) along the $\tilde{x}$-axis.

Figure 11

This is similar to Fig. 10 but for the case $t=10$. It shows clears signs of reflected waves in the both ADM and BSSN data while the lattice data is mostly flat apart from two small bumps aligned to the wings of the BSSN bump.

Figure 12

This pair of figures record the maximum value of the Brill constraints $C_1$ and $C_4$ across the lattice for $0<t<10$. Note that the constraints remain bounded and appear to decay towards a constant but nonzero value during the evolution. The nonzero value is probably tied to the truncation error in solving the Hamiltonian constraint (80). The small bumps at approximately $t=5$ and $t=10$ in the left hand figure are probably due to reflections from the outer boundary (though this was not tested). The remaining constraints $C_2$, $C_3$ and $C_5$ are not included here as they show much the same behavior as shown above.

Figure 13

The top row of this figure shows how effective the numerical dissipation can be in suppressing the axisymmetric instabilities. The data differs only in the choice of the dissipation parameter, on the left $\epsilon =0.1$ while on the right $\epsilon =1.0$. The bottom row shows data along the $\tilde{x}$ axis for four choices of the dissipation parameter, $\epsilon =0.1$ (red), $\epsilon =0.2$ (blue), $\epsilon =0.5$ (green) and $\epsilon =1.0$ (black). The lower right figure shows that the dissipation has only a small effect on the peaks of the wave at $t=5$.

Figure 14

This figure is similar to Fig. 10 but in this case showing the evolutions of the Teukolsky data. There are no obvious boundary waves but the bump in the BSSN data remains. The lattice data again looks smooth and flat behind the main wave.

Figure 15

As per Fig. 14 but at $t=10$. The BSSN bump has grown by a about 50% over the period $t=0$ to $t=10$. There is also a very small bump in the lattice data near the origin.

Figure 16

These plots show the behavior of the $C_1$ constraint (126) for the evolution of the Teukolsky initial data. The plots in the top left (ADM), top right (BSSN) and bottom left (SLGR) show the evolution of the maximum of $C_1$ across the $xy$ plane. The colours in the ADM and BSSN plots correspond to $N_x=N_y=N_z=26$ (red), 50 (blue), 100 (green) and 200 (black) while for the lattice the corresponding numbers are 25, 51,101 and 201. The plot in bottom right shows the values of $C_1$ along the $\tilde{x}$-axis for the lattice data at $t=5$ for three lattices, $N_x=N_y=N_z=51$ (red), 101 (blue) and 201 (green).

Figure 17

This pair of plots shows the behavior the BSSN bump as a function of the number of grid points (left plot with $N_z=26$ (red), $N_z=50$ (blue), $N_z=100$ (green) and $N_z=200$ (black)) and as a function of time (right plot for $t=5$ to $t=10$ in steps of 1). The left plot shows that as the number of grid points is increased the size of the bump decreases while the right plot shows that the bump increases linearly with time. This bump is the source of the linear growth in the constraint seen in Fig. 16.

Figure 18

These plots were created by evolving two sets of initial data, one with $N_x=N_y=N_z=101$, the other with $N_x=N_y=N_z=201$. Both initial data sets used $\mathrm{\Delta }x=\mathrm{\Delta }y=\mathrm{\Delta }z=0.1$. There are two curves in the right plot, both for $R_{xyxy}$, one on the small grid (red) and the other on the larger grid (blue). Note how the red curve lies directly on top of the blue curve. The plots on the left show the difference in $R_{xyxy}$ between the two evolutions on $|x|<5$. The green curve is for the BSSN data while the red curve is for the lattice data.