A high-order shock capturing discontinuous Galerkin–finite difference hybrid method for GRMHD

Abstract

We present a discontinuous Galerkin (DG)–finite difference (FD) hybrid scheme that allows high-order shock capturing with the DG method for general relativistic magnetohydrodynamics. The hybrid method is conceptually quite simple. An unlimited DG candidate solution is computed for the next time step. If the candidate solution is inadmissible, the time step is retaken using robust FD methods. Because of its a posteriori nature, the hybrid scheme inherits the best properties of both methods. It is high-order with exponential convergence in smooth regions, while robustly handling discontinuities. We give a detailed description of how we transfer the solution between the DG and FD solvers, and the troubled-cell indicators necessary to robustly handle slow-moving discontinuities and simulate magnetized neutron stars. We demonstrate the efficacy of the proposed method using a suite of standard and very challenging 1D, 2D, and 3D relativistic magnetohydrodynamics test problems. The hybrid scheme is designed from the ground up to efficiently simulate astrophysical problems such as the inspiral, coalescence, and merger of two neutron stars.

A positivity-preserving adaptive-order finite-difference scheme for GRMHD

“…We plan on adopting the flux limiter of [13,22] to arrive at a scheme that is positivitypreserving for both the reconstruction and the time integration. Additionally, we have combined the PPAO scheme with our discontinuous Galerkin-finite difference hybrid scheme [32] and are working towards using the PPAO scheme to evolve the spacetime together with the GRMHD equations. We are also adding support for moving and semi-unstructured meshes, like cubed-sphere domains.…”

“…We perform simulations of both magnetized and non-magnetized TOV stars. We adopt the same configuration as in [32,45,70]. Specifically, we use a polytropic equation of state,…”

“…the spacetime is static. We use a cube [−13, 13] 3 in geometric units, which is slightly higher resolution than was used in [32,45,70]. We convert grid spacing to meters assuming a maximum mass of the neutron star of M max = 2M ⊙ .…”

“…As a final challenging 3d test case, we simulate a uniformly rotating neutron star with a ratio of polar to equatorial radii of 0.7, as in [32,45] and similar to [73]. The initial data is constructed using the RotNS code described in [74,75].…”

“…Finally, in section 3 we show results from a large variety of standard and difficult test problems in 1d, 2d, and 3d GRMHD. In appendix we briefly comment on how the scheme presented here can be used in a discontinuous Galerkin-finite difference hybrid method as described in [32].…”