Exponential methods for anisotropic diffusion

TL;DR

This paper introduces exponential numerical methods for solving the anisotropic diffusion equation in cosmic ray transport, enabling larger time steps and faster, highly accurate simulations compared to traditional implicit methods.

Contribution

It proposes exponential matrix exponential methods for anisotropic diffusion, allowing larger time steps and improved efficiency in cosmic ray transport simulations.

Findings

Exponential methods enable larger time steps, sometimes as large as the entire simulation time.

Achieve high accuracy with l2 error ≤ 10^{-10}.

Outperform traditional implicit integrators in speed and efficiency.

Abstract

The anisotropic diffusion equation is imperative in understanding cosmic ray diffusion across the Galaxy, the heliosphere, and its interplay with the ambient 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 diffusion 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 homogeneous linear problems, is unable to improve on the exponential-based methods that directly evaluate the matrix exponential.