Abstract
The Relativistic Resistive Magnetohydrodynamic (RRMHD) equations are a hyperbolic system of partial differential equations used to describe the dynamics of relativistic magnetized fluids with a finite conductivity. Close to the ideal magnetohydrodynamic regime, the source term proportional to the conductivity becomes potentially stiff and cannot be handled with standard explicit time integration methods. We propose a new class of methods to deal with the stiffness fo the system, which we name Minimally Implicit Runge-Kutta methods. These methods avoid the development of numerical instabilities without increasing the computational costs in comparison with explicit methods, need no iterative extra loop in order to recover the primitive (physical) variables, the analytical inversion of the implicit operator is trivial and the several stages can actually be viewed as stages of explicit Runge-Kutta methods with an effective time-step. We test these methods with two different one-dimensional test beds in varied conductivity regimes, and show that our second-order schemes satisfy the theoretical expectations.
Highlights
The Relativistic Resistive Magnetohydrodynamic (RRMHD) equations are a hyperbolic system of partial differential equations used to describe the fluid dynamics in the presence of magnetic fields with a finite conductivity, and when the velocities involved are close to speed of light, c, or the fluid specific energies become comparable or larger than c2
In addition to these conserved variables, Komissarov [1] has extended the system by two extra scalar potentials which control the evolution of the solenoidal constraint for the magnetic field and the divergence of the electric field following the ideas of Dedner et al [2]
The Minimally Implicit RK (MIRK) methods reduce to the optimal TVD explicit RK methods of Shu & Osher [8] for the SEj and SY operators, and implicitly evolve the stiff source terms as we explain in section; the proposed strategy needs no iterative loop over the electric field, the analytical inversion of the implicit operator is trivial and the several stages can be viewed as stages from explicit
Summary
The Relativistic Resistive Magnetohydrodynamic (RRMHD) equations are a hyperbolic system of partial differential equations used to describe the fluid dynamics in the presence of magnetic fields with a finite (but potentially large) conductivity, and when the velocities involved are close to speed of light, c, or the fluid specific energies become comparable or larger than c2. The RRMHD system of equations can be written as a set of evolution equations corresponding to the conservation of rest-mass, momentum, energy, magnetic flux and electric charge. In addition to these conserved variables, Komissarov [1] has extended the system by two extra scalar potentials which control the evolution of the solenoidal constraint for the magnetic field and the divergence of the electric field following the ideas of Dedner et al [2].
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