Simulating magnetized neutron stars with discontinuous Galerkin methods

Discontinuous Galerkin methods are popular because they can achieve high order where the solution is smooth, because they can capture shocks while needing only nearest-neighbor communication, and because they are relatively easy to formulate on complex meshes. We perform a detailed comparison of various limiting strategies presented in the literature applied to the equations of general relativistic magnetohydrodynamics. We compare the standard $\mathrm{minmod}/\mathrm{\Lambda }{\mathrm{\Pi }}^N$ limiter, the hierarchical limiter of Krivodonova, the simple WENO limiter, the HWENO limiter, and a discontinuous Galerkin-finite-difference hybrid method. The ultimate goal is to understand what limiting strategies are able to robustly simulate magnetized Tolman-Oppenheimer-Volkoff stars without any fine-tuning of parameters. Among the limiters explored here, the only limiting strategy we can endorse is a discontinuous Galerkin-finite-difference hybrid method.

Figure 1

A comparison of different limiters used to stabilize the evolution of the Riemann problem 1 of [66]. The problem is solved using 128 third-order (${\mathrm{P}}_2$) elements. The DG-FD hybrid scheme significantly outperforms the other limiters both in robustness and accuracy.

Figure 2

The difference between the analytic and numerical solution of the Riemann problem 1 of [66] at $t=0.4$ for the DG-FD ${\mathrm{P}}_2$ scheme (solid light blue curve) and the DG-FD ${\mathrm{P}}_5$ scheme (dashed purple curve). The ${\mathrm{P}}_5$ scheme is able to resolve the discontinuities just as well as the ${\mathrm{P}}_2$ scheme, while also admitting fewer unphysical oscillations away from the discontinuities.

Figure 3

Cylindrical blast wave $\rho$ at $t=4$ comparing the DG-FD hybrid scheme, the $\mathrm{\Lambda }{\mathrm{\Pi }}^N$, Krivodonova, simple WENO, and HWENO limiters using ${\mathrm{P}}_2$ DG, as well as the DG-FD scheme using ${\mathrm{P}}_5$ DG. There are 192 degrees of freedom per dimension, comparable to what is used when testing FD schemes. We see that only the DG-FD hybrid scheme really resolves the features to an acceptable level, and the $\mathrm{\Lambda }{\mathrm{\Pi }}^N$ and Krivodonova smear out the solution almost completely. In the plots of the DG-FD hybrid scheme the regions surrounded by black squares have switched from DG to FD at the final time.

Figure 4

Magnetic rotor $\rho$ at $t=0.4$ comparing the DG-FD hybrid scheme, the $\mathrm{\Lambda }{\mathrm{\Pi }}^N$, Krivodonova, simple WENO, and HWENO limiters using ${\mathrm{P}}_2$ DG, as well as the DG-FD scheme using ${\mathrm{P}}_5$ DG. There are 192 degrees of freedom per dimension, comparable to what is used when testing FD schemes. We see that only the DG-FD hybrid scheme really resolves the features to an acceptable level, and the $\mathrm{\Lambda }{\mathrm{\Pi }}^N$ and Krivodonova smear out the solution almost completely. The simple WENO limiter fails to solve the problem. In the plots of the DG-FD hybrid scheme the regions surrounded by black squares have switched from DG to FD at the final time.

Figure 5

$B^x$ for the magnetic loop advection problem. The left half of each plot is at the initial time, while the right half is after one period ($t=2.4$). We compare the DG-FD hybrid scheme, the $\mathrm{\Lambda }{\mathrm{\Pi }}^N$, Krivodonova, simple WENO, and HWENO limiters using ${\mathrm{P}}_2$ DG, as well as the DG-FD scheme using ${\mathrm{P}}_5$ DG. There are 192 degrees of freedom per dimension, comparable to what is used when testing FD schemes. In the plots of the DG-FD hybrid scheme the regions surrounded by black squares have switched from DG to FD at the final time.

Figure 6

The divergence cleaning field $\mathrm{\Phi }$ for the magnetic loop advection problem after one period ($t=2.4$) comparing the DG-FD hybrid scheme, the $\mathrm{\Lambda }{\mathrm{\Pi }}^N$, Krivodonova, simple WENO, and HWENO limiters using ${\mathrm{P}}_2$ DG, as well as the DG-FD scheme using ${\mathrm{P}}_5$ DG. There are 192 degrees of freedom per dimension, comparable to what is used when testing FD schemes. In the plots from the DG-FD hybrid scheme the regions surrounded by black squares have switched from DG to FD at the final time.

Figure 7

Magnetized Kelvin-Helmholtz instability $\rho$ at $t=1.6$ comparing the DG-FD hybrid scheme, the $\mathrm{\Lambda }{\mathrm{\Pi }}^N$, Krivodonova, simple WENO, and HWENO limiters using ${\mathrm{P}}_2$ DG, as well as the DG-FD scheme using ${\mathrm{P}}_5$ DG. There are 192 degrees of freedom per dimension, comparable to what is used when testing FD schemes. Only the DG-FD hybrid scheme and the HWENO limiter produce reasonable results, while the $\mathrm{\Lambda }{\mathrm{\Pi }}^N$ limiter has very low effective resolution, and the Krivodonova and simple WENO limiters smear out the solution almost completely. In the plots of the DG-FD hybrid scheme the regions surrounded by black squares have switched from DG to FD at the final time.

Figure 8

The maximum density over the grid $\max (\rho )$ divided by the maximum density over the grid at $t=0$ for three different resolutions for the nonmagnetized TOV star simulations. The six-element simulation uses FD throughout the interior of the star, while 12- and 24-element simulations use DG. The increased high-frequency content in 12- and 24-element simulations occurs because the high-order DG scheme is able to resolve higher oscillation modes in the star. The maximum density in the six-element case drifts down at early times because of the low resolution and the relatively low accuracy of using FD at the center.

Figure 9

The power spectrum of the maximum density for three different resolutions for the nonmagnetized TOV star simulations. The six-element simulation uses FD throughout the interior of the star, while the 12- and 24-element simulations use DG. When the high-order DG scheme is used, more oscillation frequencies are resolved. The vertical dashed lines correspond to the known frequencies in the Cowling approximation [82].

Figure 10

The maximum density over the grid $\max (\rho )$ divided by the maximum density over the grid at $t=0$ for the best two cases using classical limiters for the nonmagnetized TOV star simulations. The HWENO limiter is only stable for a ${\mathrm{P}}_2$ DG solver. Simple WENO (not plotted) gives similar results. The Krivodonova limiter only succeeded at some resolutions (three of the 16 attempted runs) and the shown best result is noticeably noisier than the subcell limiter.

Figure 11

The maximum density over the grid $\max (\rho )$ divided by the maximum density over the grid at $t=0$ for three different resolutions for the magnetized TOV star simulation. The six-element simulation uses FD throughout the interior of the star, while 12- and 24-element simulations use DG. The increased high-frequency content in 12- and 24-element simulations occurs because the high-order DG scheme is able to resolve higher oscillation modes in the star. The maximum density in the six-element case drifts down at early times because of the low resolution and the relatively low accuracy of using FD at the center.

Figure 12

The power spectrum of the maximum density for three different resolutions of the magnetized TOV star simulations. The six-element simulation uses FD throughout the interior of the star, while the 12- and 24-element simulations use DG. When the high-order DG scheme is used, more oscillation frequencies are resolved. The vertical dashed lines correspond to the known frequencies in the Cowling approximation.