von Neumann Stability Analysis of Braginskii MHD Thermal Conduction, Viscosity, Magnetic Diffusion, and Thermoelectric Effects and an Implementation in Athena++

Athena++ is a finite-volume astrophysical magnetohydrodynamics (MHD) code. As of writing, isotropic thermal conduction and kinematic viscosity, Ohmic resistivity, and ambipolar diffusion are present in the main branch. Here we present a fork of Athena++, ``Athena-Aniso''. It implements fully anisotropic thermal conduction (parallel, perpendicular, and the Righi--Leduc effect); anisotropic viscosity (all five Braginskii viscosity tensors); the Hall effect; and several thermoelectric effects (Peltier, Seebeck, Nernst, and Ettingshausen). These effects can be weighted by ionisation fraction in a partially ionised gas, with the neutral part diffusing isotropically. We have also added isotropic bulk viscosity. The Biermann battery is not implemented here. The dissipative terms have super-time-stepping support, while the non-dissipative terms are combined with the ideal MHD flux. A CFL condition for all terms is derived and implemented using von Neumann stability analysis. The dissipative terms are found to be stable under first-order time integration. The non-dissipative terms are found to be stable under third-order time integration. We additionally derive new signal speeds for a modified HLLE Riemann solver. These suppress grid-scale non-linear instabilities still present in the non-dissipative terms. The Nernst/Ettingshausen system is found to be ill-posed unless it is stabilised by both perpendicular thermal conduction and perpendicular magnetic diffusion. We derive a well-posedness criterion for this. We provide numerical tests of each implemented effect against derived analytical solutions.