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 o
f 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 5th 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.
In this paper we present a novel arbitrary high order accurate discontinuous Galerkin (DG) finite element method on space-time adaptive Cartesian meshes (AMR) for hyperbolic conservation laws in multiple space dimensions, using a high order aposterio
ri sub-cell ADER-WENO finite volume emph{limiter}. Notoriously, the original DG method produces strong oscillations in the presence of discontinuous solutions and several types of limiters have been introduced over the years to cope with this problem. Following the innovative idea recently proposed in cite{Dumbser2014}, the discrete solution within the troubled cells is textit{recomputed} by scattering the DG polynomial at the previous time step onto a suitable number of sub-cells along each direction. Relying on the robustness of classical finite volume WENO schemes, the sub-cell averages are recomputed and then gathered back into the DG polynomials over the main grid. In this paper this approach is implemented for the first time within a space-time adaptive AMR framework in two and three space dimensions, after assuring the proper averaging and projection between sub-cells that belong to different levels of refinement. The combination of the sub-cell resolution with the advantages of AMR allows for an unprecedented ability in resolving even the finest details in the dynamics of the fluid. The spectacular resolution properties of the new scheme have been shown through a wide number of test cases performed in two and in three space dimensions, both for the Euler equations of compressible gas dynamics and for the magnetohydrodynamics (MHD) equations.
We study the Richtmyer--Meshkov (RM) instability of a relativistic perfect fluid by means of high order numerical simulations with adaptive mesh refinement (AMR). The numerical scheme adopts a finite volume Weighted Essentially Non-Oscillatory (WENO)
reconstruction to increase accuracy in space, a local space-time discontinuous Galerkin predictor method to obtain high order of accuracy in time and a high order one-step time update scheme together with a cell-by-cell space-time AMR strategy with time-accurate local time stepping. In this way, third order accurate (both in space and in time) numerical simulations of the RM instability are performed, spanning a wide parameter space. We present results both for the case in which a light fluid penetrates into a higher density one (Atwood number $A>0$), and for the case in which a heavy fluid penetrates into a lower density one (Atwood number $A<0$). We find that, for large Lorentz factors gamma_s of the incident shock wave, the relativistic RM instability is substantially weakened and ultimately suppressed. More specifically, the growth rate of the RM instability in the linear phase has a local maximum which occurs at a critical value of gamma_s ~ [1.2,2]. Moreover, we have also revealed a genuine relativistic effect, absent in Newtonian hydrodynamics, which arises in three dimensional configurations with a non-zero velocity component tangent to the incident shock front. In this case, the RM instability is strongly affected, typically resulting in less efficient mixing of the fluid.
We present a novel cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on unstructured triangular meshes that is high order accurate in space and time and that also allows for time-accurate local time stepping (LTS). The new
scheme uses the following basic ingredients: a high order WENO reconstruction in space on unstructured meshes, an element-local high-order accurate space-time Galerkin predictor that performs the time evolution of the reconstructed polynomials within each element, the computation of numerical ALE fluxes at the moving element interfaces through approximate Riemann solvers, and a one-step finite volume scheme for the time update which is directly based on the integral form of the conservation equations in space-time. The inclusion of the LTS algorithm requires a number of crucial extensions, such as a proper scheduling criterion for the time update of each element and for each node; a virtual projection of the elements contained in the reconstruction stencils of the element that has to perform the WENO reconstruction; and the proper computation of the fluxes through the space-time boundary surfaces that will inevitably contain hanging nodes in time due to the LTS algorithm. We have validated our new unstructured Lagrangian LTS approach over a wide sample of test cases solving the Euler equations of compressible gasdynamics in two space dimensions, including shock tube problems, cylindrical explosion problems, as well as specific tests typically adopted in Lagrangian calculations, such as the Kidder and the Saltzman problem. When compared to the traditional global time stepping (GTS) method, the newly proposed LTS algorithm allows to reduce the number of element updates in a given simulation by a factor that may depend on the complexity of the dynamics, but which can be as large as 4.7.
We present the results of two-dimensional magnetohydrodynamical numerical simulations of relativistic magnetic reconnection, with particular emphasis on the dynamics of Petschek-type configurations with high Lundquist numbers, S ~ 10^5-10^8. The nume
rical scheme adopted, allowing for unprecedented accuracy for this type of calculations, is based on high order finite volume and discontinuous Galerkin methods as recently proposed by Dumbser & Zanotti (2009). The possibility of producing high Lorentz factors is discussed, by studying the effects produced on the dynamics by different magnetization and resistivity regimes. We show that Lorentz factors close to ~4 can be produced for a plasma magnetization parameter sigma=20. Moreover, we find that the Sweet-Parker layers are unstable, generating secondary magnetic islands, but only for S>S_c~10^8, much larger than what is reported in the Newtonian regime.
We present the results of two-dimensional and three-dimensional magnetohydrodynamical numerical simulations of relativistic magnetic reconnection, with particular emphasis on the dynamics of the plasma in a Petschek-type configuration with high Lundq
uist numbers, Ssim 10^5-10^8. The numerical scheme adopted, allowing for unprecedented accuracy for this type of calculations, is based on high order finite volume and discontinuous Galerkin methods as recently proposed by citet{Dumbser2009}. The possibility of producing high Lorentz factors is discussed, showing that Lorentz factors close to sim 4 can be produced for a plasma parameter sigma_m=20. Moreover, we find that the Sweet-Parker layers are unstable, generating secondary magnetic islands, but only for S > S_c = 10^8, much larger than what is reported in the Newtonian regime. Finally, the effects of a mildly anisotropic Ohm law are considered in a configuration with a guide magnetic field. Such effects produce only slightly faster reconnection rates and Lorentz factors of about 1% larger with respect to the perfectly isotropic Ohm law.
In this paper we propose the first better than second order accurate method in space and time for the numerical solution of the resistive relativistic magnetohydrodynamics (RRMHD) equations on unstructured meshes in multiple space dimensions. The non
linear system under consideration is purely hyperbolic and contains a source term, the one for the evolution of the electric field, that becomes stiff for low values of the resistivity. For the spatial discretization we propose to use high order $PNM$ schemes as introduced in cite{Dumbser2008} for hyperbolic conservation laws and a high order accurate unsplit time discretization is achieved using the element-local space-time discontinuous Galerkin approach proposed in cite{DumbserEnauxToro} for one-dimensional balance laws with stiff source terms. The divergence free character of the magnetic field is accounted for through the divergence cleaning procedure of Dedner et al. cite{Dedneretal}. To validate our high order method we first solve some numerical test cases for which exact analytical reference solutions are known and we also show numerical convergence studies in the stiff limit of the RRMHD equations using $PNM$ schemes from third to fifth order of accuracy in space and time. We also present some applications with shock waves such as a classical shock tube problem with different values for the conductivity as well as a relativistic MHD rotor problem and the relativistic equivalent of the Orszag-Tang vortex problem. We have verified that the proposed method can handle equally well the resistive regime and the stiff limit of ideal relativistic MHD. For these reasons it provides a powerful tool for relativistic astrophysical simulations involving the appearance of magnetic reconnection.
