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

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 (∆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.

Braginskii viscosity on an unstructured, moving mesh accelerated with super-time-stepping

“…11. We find, in agreement with expectations (Vaidya et al 2017), that the number of total steps scales as N 3/2…”

“…We perform six simulations with increasing spatial resolu- tion. Following the procedure used in Vaidya et al (2017), we set the ratio of super time step to grid spacing, τ/∆x, to be constant. The resulting number of STS stages then increase in increments of 2 from s = 3 at the lowest resolution to s = 13 at the highest.…”

“…This implementation is based on Meyer et al (2012); Meyer et al (2014) which describe a second-order accurate Runge-Kutta Legendre method (RKL2). The RKL2 method has been implemented in the Pluto code (Vaidya et al 2017) for anisotropic heat conduction but has not previously been used for anisotropic viscosity. We briefly outline the merits of STS and how we have applied it to Braginskii viscosity.…”

Modeling Star–Planet Interactions in Far-out Planetary and Exoplanetary Systems

“…A constant value of magnetic resistivity η is used in our model as the causal mechanism for non-ideal processes like magnetic reconnections that are prevalent in regions of interaction of winds with the planetary atmosphere. The super-time-stepping scheme included in the code is implemented for this purpose (Vaidya et al 2017). Figure 1.…”

A Comparison Study of Extrapolation Models and Empirical Relations in Forecasting Solar Wind

“…Further, PLUTO code supports adaptive mesh refinement and various non-ideal MHD processes including magnetic resistivity (Mignone et al, 2012) and Hall-MHD. The code also has support for anisotropic thermal conduction (Vaidya et al, 2017) and optical thin cooling. In last 5 years, problems pertaining to solar and magnetospheric physics have also been tackled with PLUTO code (Reale et al, 2016;Sarkar et al, 2017;Bharati Das et al, 2019;Réville et al, 2020).…”