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A resistive MHD module in the GPU-accelerated GRMHD code GRaM-X

Sara Azizi, Swapnil Shankar, Philipp Mösta, Roland Haas, Erik Schnetter

TL;DR

The work addresses the need to model non-ideal relativistic plasmas in strong-gravity environments by introducing a resistive GRMHD module into the GPU-accelerated GRM-X code. It implements an IMEX-RK2 scheme with a 1D primitive-recovery strategy, coupling Maxwell's equations to GRMHD via a stiff Ohm's law and evolving with a Z4c spacetime formulation. The module is validated through a comprehensive suite of 1D, 2D, and 3D tests, showing correct convergence to the ideal-MHD limit at high conductivity and accurate resistive behavior in finite-conductivity regimes, including both confined and extended magnetic field configurations around TOV stars. This advancement enables efficient large-scale simulations of mergers, accretion, and jets in multi-messenger astrophysics, with planned improvements in microphysical EoS and more robust primitive solvers for broader physical fidelity.

Abstract

Relativistic macroscopic plasma dynamics can be described by general-relativistic magnetohydrodynamics. In many high-energy astrophysical settings, such as the interior dynamics of magnetized stars, the ideal GRMHD approximation, in which we assume infinite conductivity, provides an excellent description. However, ideal GRMHD neglects resistive effects that are essential for processes such as magnetic reconnection, dissipation, and magnetospheric dynamics. Incorporating resistivity into astrophysical plasma models accounts for the fact that plasmas in such environments are not perfect conductors. We present a resistive version of the GPU-accelerated GRMHD code GRaM-X, which evolves the full resistive GRMHD equations using the Z4c formalism for Einstein's equations. We implement a second-order implicit-explicit Runge-Kutta scheme to handle stiff source terms, obtain the primitive quantities from the conserved quantities using a one-dimensional recovery method, and employ the HLLE Riemann solver in combination with TVD and WENO reconstruction schemes. We validate the module using a range of standard tests, including 1D shocktubes, current sheets, Alfvén waves, 2D cylindrical explosions, and 3D TOV stars. The results of these tests demonstrate accurate recovery of the ideal MHD limit, correct resistive behavior, and stable evolution in dynamical spacetimes. Leveraging the GPU-accelerated resistive version of GRaM-X enables efficient large-scale simulations, paving the way for realistic studies of binary mergers, accretion flows, and relativistic jets within the framework of multi-messenger astrophysics.

A resistive MHD module in the GPU-accelerated GRMHD code GRaM-X

TL;DR

The work addresses the need to model non-ideal relativistic plasmas in strong-gravity environments by introducing a resistive GRMHD module into the GPU-accelerated GRM-X code. It implements an IMEX-RK2 scheme with a 1D primitive-recovery strategy, coupling Maxwell's equations to GRMHD via a stiff Ohm's law and evolving with a Z4c spacetime formulation. The module is validated through a comprehensive suite of 1D, 2D, and 3D tests, showing correct convergence to the ideal-MHD limit at high conductivity and accurate resistive behavior in finite-conductivity regimes, including both confined and extended magnetic field configurations around TOV stars. This advancement enables efficient large-scale simulations of mergers, accretion, and jets in multi-messenger astrophysics, with planned improvements in microphysical EoS and more robust primitive solvers for broader physical fidelity.

Abstract

Relativistic macroscopic plasma dynamics can be described by general-relativistic magnetohydrodynamics. In many high-energy astrophysical settings, such as the interior dynamics of magnetized stars, the ideal GRMHD approximation, in which we assume infinite conductivity, provides an excellent description. However, ideal GRMHD neglects resistive effects that are essential for processes such as magnetic reconnection, dissipation, and magnetospheric dynamics. Incorporating resistivity into astrophysical plasma models accounts for the fact that plasmas in such environments are not perfect conductors. We present a resistive version of the GPU-accelerated GRMHD code GRaM-X, which evolves the full resistive GRMHD equations using the Z4c formalism for Einstein's equations. We implement a second-order implicit-explicit Runge-Kutta scheme to handle stiff source terms, obtain the primitive quantities from the conserved quantities using a one-dimensional recovery method, and employ the HLLE Riemann solver in combination with TVD and WENO reconstruction schemes. We validate the module using a range of standard tests, including 1D shocktubes, current sheets, Alfvén waves, 2D cylindrical explosions, and 3D TOV stars. The results of these tests demonstrate accurate recovery of the ideal MHD limit, correct resistive behavior, and stable evolution in dynamical spacetimes. Leveraging the GPU-accelerated resistive version of GRaM-X enables efficient large-scale simulations, paving the way for realistic studies of binary mergers, accretion flows, and relativistic jets within the framework of multi-messenger astrophysics.
Paper Structure (18 sections, 46 equations, 10 figures, 1 table)

This paper contains 18 sections, 46 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Evolution equations
  3. 3+1 Decomposition of General Relativity
  4. Maxwell's Equations
  5. Magnetohydrodynamics Equations
  6. Resistive GRMHD System
  7. Numerical methods
  8. Implicit-Explicit Runge-Kutta Methods
  9. Recovery of Primitive Variables from Conserved Variables
  10. Numerical Tests
  11. 1D MHD Shocktube
  12. 2D Cylindrical Explosion
  13. Self-similar Current Sheet
  14. Large Amplitude Circularly Polarized Alfvén Waves
  15. 3D TOV Star
  16. ...and 3 more sections

Figures (10)

  • Figure 1: Left panel: Magnetic field component $B^y$ as a function of $x$ for the shock-tube problem. The figure shows three lines, each corresponding to a different resolution at $t = 0.4$. The conductivity is uniform, with a value of $\sigma_0 = 10^6$. Right panel: Magnetic field component $B^y$ as a function of $x$ for varying uniform conductivities. As the conductivity rises, the solution approaches the ideal MHD solution. It is important to observe that when $\sigma_0 =0$, the solution represents a discontinuity traveling at the speed of light, which corresponds to Maxwell's equations in a vacuum.
  • Figure 2: Left panel: The evolution of a non-uniform conductivity $\sigma$ in the shocktube problem for varying values of $\gamma$. Right panel: The magnetic field component $B^y$ as a function of $x$ for various values of $\gamma$ at $t = 0.4$. A fixed value of $\sigma_0 = 10^6$ is used in all cases.
  • Figure 3: The magnetic field components $B^x$ in the left panel and $B^y$ in the right panel at time $t = 4$ are depicted in this figure. A constant conductivity of $\sigma = 10^6$ is used.
  • Figure 4: This figure shows the magnetic field component $B^y$ as a function of $x$. Here, the conductivity is uniform, with a value of $\sigma = 10^2$, and the solution is calculated and displayed at $t = 1$ (initial time) and $t = 10$ with resolution $\Delta x = 0.005$. The initial time is set to $t = 1$ (rather than $t=0$) to avoid a singularity in the analytic solution.
  • Figure 5: The $B^y$ component of the magnetic field for three different spatial resolutions $\Delta x = \{1/50, 1/100, 1/200\}$, along with the exact solution depicted by the solid black line.
  • ...and 5 more figures