An implicit scheme for solving the anisotropic diffusion of heat and cosmic rays in the RAMSES code

Astrophysical plasmas are subject to a tight connection between magnetic fields and the diffusion of particles, which leads to an anisotropic transport of energy. Under the fluid assumption, this effect can be reduced to an advection-diffusion equation augmenting the equations of magnetohydrodynamics. We introduce a new method for solving the anisotropic diffusion equation using an implicit finite-volume method with adaptive mesh refinement and adaptive time-stepping in the RAMSES code. We apply this numerical solver to the diffusion of cosmic ray energy, and diffusion of heat carried by electrons, which couple to the ion temperature. We test this new implementation against several numerical experiments and apply it to a simple supernova explosion with a uniform magnetic field.