Exponential methods for anisotropic diffusion

Abstract

The anisotropic diffusion equation is of crucial importance in understanding cosmic ray 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 ≤ 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.

Efficient numerical methods for Anisotropic Diffusion of Galactic Cosmic Rays

“…In the near future, we will append LeXInt to Picard to study CR transport in the Galaxy using the exponential quadrature methods in combination with the Leja scheme. Yet another approach would be to penalise the anisotropic diffusion term [4] -the stiff terms would be treated implicitly whilst the non-stiff term would be solved explicitly. We will also consider continuous energy losses as well as reacceleration whilst investigating the impact of anisotropic CR diffusion in the Galactic magnetic field using Picard.…”

LeXInt: Package for exponential integrators employing Leja interpolation

“…The method of polynomial interpolation at Leja points [30,31,32] to compute the matrix exponential and the ϕ l (z) functions was proposed in [19,20], and has subsequently been shown to be highly competitive with the traditionallyused Krylov-based methods [19,23,33]. This method has been described theoretically in great detail in [19,34], and we have described the working algorithm in our previous works [35,23,24,36,33]. To avoid repetition, we simply state the equation to compute the polynomial:…”