Efficient implementation of finite volume methods in numerical relativity

Centered finite volume methods are considered in the context of numerical relativity. A specific formulation is presented, in which third-order space accuracy is reached by using a piecewise-linear reconstruction. This formulation can be interpreted as an “adaptive viscosity” modification of centered finite difference algorithms. These points are fully confirmed by one-dimensional black hole simulations. In the three-dimensional case, evidence is found that the use of a conformal decomposition is a key ingredient for the robustness of black hole numerical codes.

Figure 1

Piecewise-linear reconstruction of a given function. Numerical discontinuities appear at every cell interface (dotted lines) between the left and right values (arrows and dots, respectively). Note that the original function was monotonically decreasing: all the slopes are negative. However, both the left interface values (at $i+3/2$) and the right interface ones (at $i-3/2$) show local extremes that break the monotonicity of the original function.

Figure 2

Long-term FV simulation of a 1D black hole, with a single mesh of 120 gridpoints. The evolution of the lapse is shown up to 1000 m, in intervals of 50 m (solid lines). The dotted lines correspond to 1 m, 3 m, 5 m and 25 m. Note that the plots tend to cumulate at the end, due to the exponential character of the grid, as given by (18). No slope limiters have been used in this simulation.

Figure 3

The evolution of the propagation speed is shown up to 100 m, in intervals of 10 m, for the same simulation as in Fig. 2. The maximum values are clearly seen to decrease in time. Note the exponentially decreasing tail, as a result of the choice of the radial coordinate.

Figure 4

Time evolution of the error in the mass function (logarithm of the $L_2$ norm) for three different numerical algorithms. The strictly fourth-order FD method, without extra dissipation terms, is the most accurate as expected, but crashes after a short time (measured in units of $m$). The other two algorithms (third-order accurate) get similar errors at early times, but the FV one performs much better in the long term than the FD with standard Kreiss-Oliger dissipation. The dissipation coefficient has been taken as low as allowed by code stability (see the text). All simulations were obtained with a single mesh of 120 gridpoints and using the $1+\log $ slicing prescription.

Figure 5

Local convergence evolution for the mass function in a 1D black hole simulation. We can see the predicted third-order accuracy, when using the proposed slopes (16), around $t=10\mathrm{m}$ (solid line). Note also the second-order accuracy at the boundary ($\eta =8$). At $t=100\mathrm{m}$ (dashed line), we can see the downgrade in the regions around the collapse front (the apparent horizon position is marked with a circle). As the collapse front propagates (dotted line, corresponding to $t=400\mathrm{m}$), we can see the growth of the affected regions, especially the one behind the front.

Figure 6

Plot of the trace of the extrinsic curvature at $t=12\mathrm{m}$ for a low-resolution simulation. The dotted line corresponds to the trace obtained by contraction from the individual components $K_{ij}$. The solid line is the same quantity computed directly as a primitive variable. The big difference corresponds to the transition between the collapsed and uncollapsed regions, where the lapse shows a steep profile.