  Based on a 14-moment closure for non-resistive (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 parameterized 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.
