A constrained transport method for the solution of the resistive relativistic MHD equations

Abstract

We describe a novel Godunov-type numerical method for solving the equations of resistive relativistic magnetohydrodynamics. In the proposed approach, the spatial components of both magnetic and electric fields are located at zone interfaces and are evolved using the constrained transport formalism. Direct application of Stokes' theorem to Faraday's and Ampere's laws ensures that the resulting discretization is divergence-free for the magnetic field and charge-conserving for the electric field. Hydrodynamic variables retain, instead, the usual zone-centred representation commonly adopted in finite-volume schemes. Temporal discretization is based on Runge-Kutta implicit-explicit (IMEX) schemes in order to resolve the temporal scale disparity introduced by the stiff source term in Ampere's law. The implicit step is accomplished by means of an improved and more efficient Newton-Broyden multidimensional rootfinding algorithm. The explicit step relies on a multidimensional Riemann solver to compute the line-averaged electric and magnetic fields at zone edges and it employs a one-dimensional Riemann solver at zone interfaces to update zone-centred hydrodynamic quantities. For the latter, we introduce a five-wave solver based on the frozen limit of the relaxation system whereby the solution to the Riemann problem can be decomposed into an outer Maxwell solver and an inner hydrodynamic solver. A number of numerical benchmarks demonstrate that our method is superior in stability and robustness to the more popular charge-conserving divergence cleaning approach where both primary electric and magnetic fields are zone-centered. In addition, the employment of a less diffusive Riemann solver noticeably improves the accuracy of the computations.

General-relativistic Resistive Magnetohydrodynamics with Robust Primitive-variable Recovery for Accretion Disk Simulations

“…With all quantities written as explicit functions of the iteration variables u, the Newton algorithm can be applied to minimize the residuals (57). We note that a similar 3D approach has been presented by Bucciantini & Del Zanna (2013) and Mignone et al (2019). 2.…”

“…This method was first proposed by Palenzuela et al (2009) for SRRMHD and later extended by Bucciantini & Del Zanna (2013) and Dionysopoulou et al (2013) to GRRMHD. Several improvements of the IMEX method for SRRMHD have been proposed recently by Mignone et al (2019).…”

“…Note that a multidimensional Riemann solver for the SRRMHD equations was recently presented by Mignone et al (2018Mignone et al ( , 2019 and Miranda-Aranguren et al (2018).…”

“…Takamoto & Inoue (2011) improved the semi-analytic approach of Komissarov (2007) with an implicit inversion method for the stiff source term, relaxing the restrictive time step for SRRMHD. Bucciantini & Del Zanna (2013) and Dionysopoulou et al (2013) applied the IMEX method to GRRMHD, incorporating the full resistive Ohm's law, and Mignone et al (2019) recently presented substantial improvements in SRRMHD. Bucciantini & Del Zanna (2013) and Palenzuela (2013) used an IMEX scheme to include Hall and dynamo effects in the GRRMHD evolution, through a generalized Ohm's law.…”

“…The designed recovery methods fully incorporate the electric field dynamics, such that highly magnetized regions around black holes and neutron stars can be accurately resolved in the resistive regime in between the electrovacuum (h  ¥) and the ideal-MHD limits (h  0). The methods are compared to the standard recovery schemes as presented by Palenzuela et al (2009) Dionysopoulou et al (2013), and Mignone et al (2019). We provide full details of the recovery procedure, such that it can be readily implemented in GRRMHD algorithms.…”