A robust fourth-order finite-difference discretization for the strongly anisotropic transport equation in magnetized plasmas

TL;DR

This paper introduces a high-order, implicit finite-difference scheme for accurately simulating strongly anisotropic heat transport in magnetized plasmas, overcoming previous limitations in numerical stability and efficiency.

Contribution

It develops a fourth-order spatial discretization with a multigrid preconditioner, enabling efficient, accurate, and scalable simulations of anisotropic heat transport in complex plasma geometries.

Findings

Achieves high spatial accuracy with low numerical pollution.

Demonstrates weak scaling of linear iterations with grid refinement.

Successfully simulates nonlinear MHD phenomena in 2D and 3D geometries.

Abstract

We propose a second-order temporally implicit, fourth-order-accurate spatial discretization scheme for the strongly anisotropic heat transport equation characteristic of hot, fusion-grade plasmas. Following [Du Toit et al., Comp. Phys. Comm., 228 (2018)], the scheme transforms mixed-derivative diffusion fluxes (which are responsible for the lack of a discrete maximum principle) into nonlinear advective fluxes, amenable to nonlinear-solver-friendly monotonicity-preserving limiters. The scheme enables accurate multi-dimensional heat transport simulations with up to seven orders of magnitude of heat-transport-coefficient anisotropies with low cross-field numerical error pollution and excellent algorithmic performance, with the number of linear iterations scaling very weakly with grid resolution and grid anisotropy, and scaling with the square-root of the implicit timestep. We propose a multigrid preconditioning strategy based on a second-order-accurate approximation that renders the scheme efficient and scalable under grid refinement. Several numerical tests are presented that display the expected spatial convergence rates and strong algorithmic performance, including fully nonlinear magnetohydrodynamics simulations of kink instabilities in a Bennett pinch in 2D helical geometry and of ITER in 3D toroidal geometry.