Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement

Abstract

We present a new numerical tool for solving the special relativistic ideal MHD equations that is based on the combination of the following three key features: (i) a one-step ADER discontinuous Galerkin (DG) scheme that allows for an arbitrary order of accuracy in both space and time, (ii) an a posteriori subcell finite volume limiter that is activated to avoid spurious oscillations at discontinuities without destroying the natural subcell resolution capabilities of the DG finite element framework and finally (iii) a space-time adaptive mesh refinement (AMR) framework with time-accurate local time-stepping.The divergence-free character of the magnetic field is instead taken into account through the so-called "divergence-cleaning" approach. The convergence of the new scheme is verified up to 5 th order in space and time and the results for a set of significant numerical tests including shock tube problems, the RMHD rotor and blast wave problems, as well as the Orszag-Tang vortex system are shown. We also consider a simple case of the relativistic Kelvin-Helmholtz instability with a magnetic field, emphasizing the potential of the new method for studying turbulent RMHD flows. We discuss the advantages of our new approach when the equations of relativistic MHD need to be solved with high accuracy within various astrophysical systems.

Solving 3D relativistic hydrodynamical problems with weighted essentially nonoscillatory discontinuous Galerkin methods

“…Another important development is a hybrid approach [35][36][37][38], which replaces troubled cells by an equidistant subgrid and applies FV shock capturing on these grids. This approach maintains the subcell resolution of FV methods, but increases the complexity of the implementation since two types of grids and special grid transfer operators are required.…”

hp-adaptive discontinuous Galerkin solver for elliptic equations in numerical relativity

“…Equation 19 contains integrals and therefore represents only a semi-discrete set of equations. In the next section we show how to make Eq.…”

“…In the next section we show how to make Eq. (19) fully discrete by using quadrature to approximate the integrals.…”

“…Discontinuous Galerkin (DG) methods [1][2][3][4][5][6] have matured into standard numerical methods to simulate a wide variety of partial differential equations. In the context of numerical relativity [7], discontinuous Galerkin methods have shown advantages for relativistic hyperbolic problems over traditional discretization methods such as finite difference, finite volume and spectral finite elements [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] by combining the best aspects of all three methods. As computers reach exa-scale power, new methods like DG are needed to tackle the biggest problems in numerical relativity such as realistic supernovae and binary neutron-star merger simulations on these very large machines [21].…”

CosmosDG: An hp-adaptive Discontinuous Galerkin Code for Hyper-resolved Relativistic MHD

“…In that work, the Einstein field equations were discretized in all four spacetime dimensions, thus treating time entirely equivalent to space. We are aware of only three other applications of DG to relativistic magnetohydrodynamics (MHD): Radice & Rezzolla (2011), Zanotti et al (2015) and Kidder et al (2017). The first two have significant restrictions, with the first being limited to one-dimensional spherical symmetry, while the second is limited to special relativity.…”