A Subluminal Relativistic Magnetohydrodynamics Scheme with ADER-WENO Predictor and Multidimensional Riemann Solver-Based Corrector
📝 Abstract
The relativistic magnetohydrodynamics (RMHD) set of equations has recently seen increased use in astrophysical computations. Even so, RMHD codes remain fragile. The reconstruction can sometimes yield superluminal velocities in certain parts of the mesh. In this paper we present a reconstruction strategy that overcomes this problem by making a single conservative to primitive transformation per cell followed by higher order WENO reconstruction on a carefully chosen set of primitives that guarantee subluminal reconstruction of the flow variables. For temporal evolution via a predictor step we also present second, third and fourth order accurate ADER methods that keep the velocity subluminal during the predictor step. The RMHD system also requires the magnetic field to be evolved in a divergence-free fashion. In the treatment of classical numerical MHD the analogous issue has seen much recent progress with the advent of multidimensional Riemann solvers. By developing multidimensional Riemann solvers for RMHD, we show that similar advances extend to RMHD. As a result, the face-centered magnetic fields can be evolved much more accurately using the edge-centered electric fields in the corrector step. Those edge-centered electric fields come from a multidimensional Riemann solver for RMHD which we present in this paper. In this paper we also develop several new test problems for RMHD. We show that RMHD vortices can be designed that propagate on the computational mesh as self-preserving structures. These RMHD vortex test problems provide a means to do truly multidimensional accuracy testing for RMHD codes. Several other stringent test problems are presented. We show the importance of resolution in certain test problems. Our tests include a demonstration that RMHD vortices are stable when they interact with shocks.
💡 Analysis
The relativistic magnetohydrodynamics (RMHD) set of equations has recently seen increased use in astrophysical computations. Even so, RMHD codes remain fragile. The reconstruction can sometimes yield superluminal velocities in certain parts of the mesh. In this paper we present a reconstruction strategy that overcomes this problem by making a single conservative to primitive transformation per cell followed by higher order WENO reconstruction on a carefully chosen set of primitives that guarantee subluminal reconstruction of the flow variables. For temporal evolution via a predictor step we also present second, third and fourth order accurate ADER methods that keep the velocity subluminal during the predictor step. The RMHD system also requires the magnetic field to be evolved in a divergence-free fashion. In the treatment of classical numerical MHD the analogous issue has seen much recent progress with the advent of multidimensional Riemann solvers. By developing multidimensional Riemann solvers for RMHD, we show that similar advances extend to RMHD. As a result, the face-centered magnetic fields can be evolved much more accurately using the edge-centered electric fields in the corrector step. Those edge-centered electric fields come from a multidimensional Riemann solver for RMHD which we present in this paper. In this paper we also develop several new test problems for RMHD. We show that RMHD vortices can be designed that propagate on the computational mesh as self-preserving structures. These RMHD vortex test problems provide a means to do truly multidimensional accuracy testing for RMHD codes. Several other stringent test problems are presented. We show the importance of resolution in certain test problems. Our tests include a demonstration that RMHD vortices are stable when they interact with shocks.
📄 Content
1
A Subluminal Relativistic Magnetohydrodynamics Scheme with ADER- WENO Predictor and Multidimensional Riemann Solver-Based Corrector By Dinshaw S. Balsara (dbalsara@nd.edu) and Jinho Kim (jkim46@nd.edu) Physics Department, University of Notre Dame Abstract
The relativistic magnetohydrodynamics (RMHD) set of equations has recently seen increased use in astrophysical computations. Even so, RMHD codes remain fragile. The reconstruction can sometimes yield superluminal velocities in certain parts of the mesh. The current generation of RMHD codes does not have any particularly good strategy for avoiding such an unphysical situation. In this paper we present a reconstruction strategy that overcomes this problem by making a single conservative to primitive transformation per cell followed by higher order WENO reconstruction on a carefully chosen set of primitives that guarantee subluminal reconstruction of the flow variables. For temporal evolution via a predictor step we also present second, third and fourth order accurate ADER methods that keep the velocity subluminal during the predictor step. The methods presented here are very general and should apply to other PDE systems where physical realizability is most easily asserted in the primitive variables.
The RMHD system also requires the magnetic field to be evolved in a divergence-free fashion. In the treatment of classical numerical MHD the analogous issue has seen much recent progress with the advent of multidimensional Riemann solvers. By developing multidimensional Riemann solvers for RMHD, we show that similar advances extend to RMHD. As a result, the face-centered magnetic fields can be evolved much more accurately using the edge-centered electric fields in the corrector step. Those edge-centered electric fields come from a multidimensional Riemann solver for RMHD which we present in this paper. The overall update results in a one-step, fully conservative scheme that is suited for AMR.
In this paper we also develop several new test problems for RMHD. We show that RMHD vortices can be designed that propagate on the computational mesh as self-preserving structures. These RMHD vortex test problems provide a means to do truly multidimensional 2
accuracy testing for RMHD codes. Several other stringent test problems are presented. We show the importance of resolution in certain test problems. Our tests include a demonstration that RMHD vortices are stable when they interact with shocks. I) Introduction
Just like non-relativistic magnetohydrodynamics (MHD), relativistic magnetohydrodynamics (RMHD) has seen a great deal of recent progress. Both MHD and RMHD are PDE systems with an involution constraint, where the magnetic field starts off divergence-free and stays so for all time. In a numerical code, keeping the magnetic field divergence-free requires the use of a staggered, Yee-type mesh. Likewise, both systems are non- convex with the result that their numerical treatment results in codes that tend to be more brittle than a traditional hydrodynamics code. Numerical RMHD suffers from the further deficiency that reconstructed velocities can become superluminal at certain locations of the mesh, thus making RMHD codes even more fragile than MHD codes. Several advances have been made in numerical RMHD (e.g. Aloy et al., [3], Komissarov [50], Balsara [7], Del Zanna et al. [31], Gammie et al. [42], Komissarov [51], Ryu et al. [62], Mignone & Bodo [56], Honkkila & Janhunen [47], Del Zanna et al. [32], Tchekhovskoy et al. [64], Mignone, Ugliano and Bodo [57], Anton et al. [4], McKinney et al. [55], Kim & Balsara [49], Etienne et al. [39]). In many of these papers, several very sophisticated one-dimensional Riemann solvers have been designed and many very nice higher order reconstruction strategies have been discussed. In other words, this is the traditional progress that needs to be made for producing higher order Godunov schemes for RMHD. The fact that such progress has been made is a very good thing because it sets the stage for the next round of progress reported here.
Despite all the excellent progress, RMHD codes remain brittle. Ensuring positivity of density or pressure can be a challenge. Fortunately, work done in ensuring the positivity of density and pressure in non-relativistic MHD (Balsara [13]) can be directly transcribed to RMHD. Consequently, ensuring the positivity of density and pressure in RMHD simulations is not such a big challenge any more. However, RMHD codes suffer from a further deficiency. They can frequently produce reconstructed velocities that might be superluminal in certain places on the mesh. This is a bigger difficulty because once the flow becomes superluminal, it is impossible for the RMHD code to recover. This problem is exacerbated by the fact that some 3
RMHD calculations require very high Lorentz factors, and an increasing Lorentz factor makes it more
This content is AI-processed based on ArXiv data.