Dissipative magnetohydrodynamics for nonresistive relativistic plasmas: An implicit second-order flux-conservative formulation with stiff relaxation

Based on a 14-moment closure for nonresistive (general-) relativistic viscous plasmas, we describe a new numerical scheme that is able to handle all first-order dissipative effects (heat conduction, bulk and shear viscosities), as well the anisotropies induced by the presence of magnetic fields. The latter is parametrized in terms of a thermal gyrofrequency or, equivalently, a thermal Larmor radius and allows to correctly capture the thermal Hall effect. By solving an extended Israel-Stewart-like system for the dissipative quantities that enforces algebraic constraints via stiff-relaxation, we are able to cast all first-order dissipative terms in flux-divergence form. This allows us to apply traditional high-resolution shock capturing methods to the equations, making the system suitable for the numerical study of highly turbulent flows. We present several numerical tests to assess the robustness of our numerical scheme in flat spacetime. The 14-moment closure can seamlessly interpolate between the highly collisional limit found in neutron star mergers, and the highly anisotropic limit of relativistic Braginskii magnetohydrodynamics appropriate for weakly collisional plasmas in black-hole accretion problems. We believe that this new formulation and numerical scheme will be useful for a broad class of relativistic magnetized flows.

Figure 1

One-dimensional blast wave problem for different choices of the bulk viscous relaxation time ${\tau }_{\mathrm{\Pi }}$ for fixed $\kappa /{\tau }_{\mathrm{\Pi }}=1$. The left panel shows the rest-mass density $\rho$ at time $t=0.4$, the middle panel the four-velocity component $u^x$, and the right panel the value of the bulk pressure $\mathrm{\Pi }$.

Figure 2

Thermal dissipation of an initial Gaussian temperature, $T$, profile. The transport coefficients $\kappa$ and ${\tau }_q$ are varied in each case. The different colors denote different times $t$ during the diffusion process.

Figure 3

Comparison of the dissipation of the top, $x=0$ of a Gaussian temperature, $T$, profile. Shown are two evolution profiles for two sets of transport coefficients $\zeta$ and ${\tau }_q$. These are compared to analytic solutions of the heat equation, Eq. (136), and the telegrapher’s equation, Eq. (137).

Figure 4

Anisotropic heat conductivity test. Starting from an initially hot inner cylinder at temperature $T_h$ surrounded by an ambient medium at temperature $T_c$ (left column), the evolution of relativistic causal heat conduction is shown for two different times in the middle and right columns. The rows correspond to various degrees of anisotropy with respect to the magnetic field (shown as white streamlines). We can see that in the most anisotropic case (bottom row), heat can only flow along the magnetic field lines. A more detailed description is given in Sec. 5c.

Figure 5

Two-dimensional Kelvin-Helmholtz test with heat conduction at $t=3.4$. Shown is the rest-mass density $\rho$. In the absence of shear viscosity secondary vortices form at higher resolutions, denoted by the number of grid points $N_y\times N_x$.

Figure 6

Two-dimensional Kelvin-Helmholtz test with heat conduction and shear viscosity at $t=3.4$. In the presence of shear viscosity the same converged vortex structure is recovered also at higher resolutions, denoted by the number of grid points $N_y\times N_x$. Shown are the rest-mass density $\rho$ (top row), the four-velocity $u^y$ along the shear layer (second row), and the shear stress tensor components ${\pi }^{xy}$ (third row) and ${\pi }^{yy}$ (bottom row).

Figure 7

Two-dimensional Kelvin-Helmholtz test with anisotropic heat conduction, shear viscosity and finite thermal gyrofrequency at $t=3.4$. the left column shows results in the extended magnetohydrodynamics (Braginskii-like) limit of the closure. The center column in addition adds a subgrid model to include kinetic effects (limiting the pressure anisotropies according to mirror and firehose instabilities) that increase collisionality. The right column corresponds to a nonresistive viscous simulation. The rows show the pressure anisotropy $|P_{\parallel }-P_{\perp }|$ relative to the magnetic field, the in-plane shear-stresses ${\pi }^{xy}$ and the norm of the in-plane heat flux $|\mathbf{q}|$.