Exponential methods for anisotropic diffusion

The anisotropic diffusion equation is of crucial importance in understanding cosmic ray (CR) diffusion across the Galaxy and its interplay with the Galactic magnetic field. This diffusion term contributes to the highly stiff nature of the CR transport equation. In order to conduct numerical simulations of time-dependent cosmic ray transport, implicit integrators have been traditionally favoured over the CFL-bound explicit integrators in order to be able to take large step sizes. We propose exponential methods that directly compute the exponential of the matrix to solve the linear anisotropic diffusion equation. These methods allow us to take even larger step sizes; in certain cases, we are able to choose a step size as large as the simulation time, i.e., only one time step. This can substantially speed-up the simulations whilst generating highly accurate solutions (l2 error $\leq 10^{-10}$). Additionally, we test an approach based on extracting a constant coefficient from the anisotropic diffusion equation where the constant coefficient term is solved implicitly or exponentially and the remainder is treated using some explicit method. We find that this approach, for linear problems, is unable to improve on the exponential-based methods that directly evaluate the matrix exponential.