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Very High Order $\PNM$ Schemes on Unstructured Meshes for the Resistive Relativistic MHD Equations
Michael Dumbser, Olindo Zanotti
TL;DR
The paper addresses accurate RRMHD computation on unstructured meshes across resistive and stiff high-conductivity regimes. It combines high-order PNPM spatial schemes with an element-local space-time DG predictor and divergence cleaning, and reports better-than-second-order simulations with nominal convergence through fifth order. The method handles both the resistive regime and the stiff limit of ideal relativistic MHD, supporting astrophysical simulations involving magnetic reconnection.
Problem
RRMHD simulations must cover low and high resistivity, but high conductivity makes the source term stiff and magnetic-reconnection physics can be missed by neglecting resistivity.
Method
The method combines high-order PNPM schemes on unstructured meshes, an element-local space-time discontinuous Galerkin predictor, and divergence cleaning.
Results
The method achieves better than second-order accuracy in space and time, reaches nominal convergence orders in stiff-limit tests including the electric field, and handles both resistive and stiff ideal-MHD regimes.
Takeaways & Limitations
High-order PNPM schemes on coarse unstructured meshes are promising for computationally demanding astrophysical simulations involving magnetic reconnection.
Takeaways & Limitations
The study considers only flat spacetimes in Cartesian coordinates and identifies full general relativity and more complex Ohm’s laws as future extensions.
Abstract
from arXiv · showhide
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 nonlinear 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.
1 Introduction
The paper targets accurate RRMHD simulation across low and high resistivity, where magnetic reconnection motivates retaining resistive effects and high conductivity creates stiffness. It combines high-order PNPM spatial discretization with an element-local space-time DG treatment and evaluates the method on unstructured meshes.
- Motivation: Magnetic reconnection applications motivate solving the full resistive relativistic MHD system rather than assuming infinite conductivity.The paper cites pulsar magnetospheres, soft gamma-ray repeaters, and extragalactic jets as relevant settings.
- Problem: A single computational tool is sought for both low- and high-resistivity regimes, although the equations become stiff at high conductivity.The conductivity-dependent stiffness appears in the nonlinear source term of the hyperbolic balance law.
- Approach: The method uses high-order PNPM schemes in space and an element-local space-time discontinuous Galerkin predictor for stiff source terms.The predictor is locally implicit and the global corrector is explicit, forming a one-step local predictor-global corrector method.
- Assumptions: The formulation assumes flat spacetime in Cartesian coordinates and uses the Lorentz-Heaviside convention with c = 1.Greek indices run over spacetime components and Latin indices over spatial components.
2 The Resistive Relativistic MHD Equations
The RRMHD model couples relativistic fluid and electromagnetic dynamics through Maxwell equations, Ohm’s law, and nonlinear stiff source terms. Divergence cleaning augments the system, while ideal-gas thermodynamics and conservative-to-primitive recovery complete the formulation.
- Relativistic formulation: The total energy-momentum tensor combines matter and electromagnetic contributions.The matter contribution uses the fluid four-velocity, enthalpy, and pressure; the electromagnetic contribution uses the electromagnetic tensor.
- Resistive dynamics: Resistive MHD explicitly evolves the electric field through Maxwell equations and closes the system with a relativistic Ohm’s law.The electric current is related to charge density, conductivity, electric field, magnetic field, and velocity.
- Divergence control: Divergence cleaning introduces scalar fields Ψ and Φ to propagate deviations in the electric- and magnetic-field divergences away from the solution.The augmented system contains fluid, electromagnetic, divergence, and total-charge equations.
- Conservative system: The conservative formulation defines relativistic mass, momentum, and total energy densities and assumes an ideal-gas equation of state.The source terms in the electric-field evolution equations can become stiff and are treated with the paper’s high-order procedure.
- Variable recovery: For an ideal-gas equation of state, primitive-variable recovery reduces to a quartic equation for the Lorentz factor Γ.The quartic is solved analytically and then refined with one or two Newton iterations to improve accuracy and robustness.
3 Numerical Method
The numerical method reconstructs high-order polynomials on unstructured triangular or tetrahedral meshes, then advances each element with a local space-time DG predictor and an explicit global corrector. The predictor couples fluxes and stiff sources locally through a nonlinear algebraic solve.
- PNPM reconstruction: The PNPM reconstruction operator raises piecewise-polynomial data from degree N to degree M ≥ N on unstructured meshes.Elements are triangles in two dimensions and tetrahedra in three dimensions.
- PNPM reconstruction: Reconstruction uses element stencils and an L2 projection solved as an overdetermined constrained least-squares system.The reconstructed polynomial is extended over the stencil and then restricted back to the target element.
- Space-time predictor: The local space-time DG method serves as a predictor for stiff balance laws, while the global corrector remains fully explicit and couples neighboring cells.The method is therefore local and implicit during prediction but explicit in the interelement update.
- Local solve: An element-local nonlinear algebraic system is solved iteratively using universal reference-element matrices and a simplified implicit-source Jacobian treatment.The source Jacobian is evaluated once per iteration at the current space-time average of the reconstructed state.
- Stiff initialization: In very stiff cases, the initial guess is improved by first solving a first-order Newton-Raphson problem and relaxing the electric field toward the σ →∞ equilibrium.The equilibrium-based electric-field initialization is used while holding the other conservative variables fixed.
4 Numerical Test Cases
The numerical tests assess high-order PNPM schemes on unstructured triangular meshes in stiff, resistive, and shock-dominated RRMHD regimes. Exact-solution comparisons, convergence studies, and multidimensional applications show the method’s accuracy across these settings.
- Large Amplitude Alfvén Wave: The large-amplitude Alfvén-wave test uses an exact periodic solution to assess accuracy in the stiff limit σ → ∞.The computational domain has periodic boundaries, and the final time t = 2.618033988 corresponds to one complete wave-advection period.
- Large Amplitude Alfvén Wave: The nominal order M + 1 is reached for all tested PNPM schemes, with P1P4 on NG = 8 outperforming P0P2 on NG = 64.Errors are measured for By in the convergence study, with σ = 10^7 except σ = 10^8 for P1P4.
- Large Amplitude Alfvén Wave: The stiff-source variable Ey also reaches the nominal order of accuracy for P0P2, P0P3, and P1P4 schemes.The results support uniform space-time accuracy in both stiff and non-stiff cases.
- Self-similar Current Sheet: The self-similar current-sheet test at σ = 100 shows excellent agreement with the exact solution for two fourth-order schemes on different meshes.P0P3 uses an equivalent one-dimensional resolution of 32 points, while P2P3 uses only 8 points.
- Shock Tube Problems: The shock-tube series compares RRMHD solutions across σ = 0, 1, 10, 10^2, 10^3, and 10^6 against the ideal-RMHD exact solution.The tests use an unstructured triangular mesh with characteristic size h = 1/400 and examine density and By cuts.
- Orszag–Tang Vortex: The resistive Orszag–Tang vortex develops complex shock-dominated structures at σ = 10^3, whereas σ = 10 produces fewer waves because of electric-resistivity diffusion.The simulations use a P0P2 scheme on an unstructured mesh with 55292 elements through t = 4.5.
5 Conclusions
The paper applies PNPM schemes with a local space-time discontinuous Galerkin predictor to stiff RRMHD equations, achieving better than second-order accuracy in space and time. The results support higher-order methods for coarse-mesh simulations of time-dependent astrophysical processes involving magnetic reconnection.
- PNPM schemes coupled with a local space-time discontinuous Galerkin predictor account for source terms that become stiff as conductivity approaches the ideal limit.The predictor is designed for the stiff source terms arising in the RRMHD equations.
- The computations are presented as the first better-than-second-order accurate simulations in space and time for the stiff limit of RRMHD.The authors report that the results favor higher-order PNPM schemes over standard second-order TVD schemes.
- High-order PNPM schemes achieve useful accuracy on very coarse meshes, making them promising for computationally demanding time-dependent problems involving astrophysical magnetic reconnection.
- The pressure-field study compares resistive relativistic Orszag-Tang vortex solutions at t = 0.5, 2.0, and 4.5 for conductivities σ = 10 and σ = 103.