# Scalable explicit implementation of anisotropic diffusion with Runge-Kutta-Legendre super-time-stepping

## TL;DR

This paper presents a second-order Runge-Kutta-Legendre super-time-stepping scheme for efficiently and accurately simulating anisotropic thermal conduction in astrophysical plasmas, outperforming previous methods in speed and precision.

## Contribution

It introduces a robust, fast, and second-order accurate explicit scheme for anisotropic diffusion, improving over existing super-time-stepping methods in stability and accuracy.

## Key findings

- RKL scheme is more accurate than first-order STS methods.
- Explicit conduction with RKL scales efficiently on large parallel architectures.
- The method effectively handles saturated thermal conduction in simulations.

## Abstract

An important ingredient in numerical modelling of high temperature magnetised astrophysical plasmas is the anisotropic transport of heat along magnetic field lines from higher to lower temperatures.Magnetohydrodynamics (MHD) typically involves solving the hyperbolic set of conservation equations along with the induction equation. Incorporating anisotropic thermal conduction requires to also treat parabolic terms arising from the diffusion operator. An explicit treatment of parabolic terms will considerably reduce the simulation time step due to its dependence on the square of the grid resolution ($\Delta x$) for stability. Although an implicit scheme relaxes the constraint on stability, it is difficult to distribute efficiently on a parallel architecture. Treating parabolic terms with accelerated super-time stepping (STS) methods has been discussed in literature but these methods suffer from poor accuracy (first order in time) and also have difficult-to-choose tuneable stability parameters. In this work we highlight a second order (in time) Runge Kutta Legendre (RKL) scheme (first described by Meyer et. al. 2012) that is robust, fast and accurate in treating parabolic terms alongside the hyperbolic conversation laws. We demonstrate its superiority over the first order super time stepping schemes with standard tests and astrophysical applications. We also show that explicit conduction is particularly robust in handling saturated thermal conduction. Parallel scaling of explicit conduction using RKL scheme is demonstrated up to more than $10^4$ processors.