A provably stable numerical method for the anisotropic diffusion equation in confined magnetic fields: Curvilinear coordinates and multi-block domains

TL;DR

This paper introduces a stable, accurate numerical method for solving the anisotropic diffusion equation in complex magnetic geometries using curvilinear coordinates and multi-block domains, with proven stability and verified convergence.

Contribution

It extends existing Cartesian-based methods to complex geometries with a multi-block approach and provides a proof of semi-discrete stability.

Findings

Method demonstrates convergence in multi-domain tests.

Stable semi-discrete implementation proven via energy estimates.

Qualitative results in complex magnetic geometries are shown.

Abstract

We present a robust and accurate numerical method for the anisotropic diffusion equation in curvilinear coordinates. This study extends the recent work [Muir et al., Computer Physics Communications, 2025] for solving the anisotropic diffusion equation in magnetic fields from Cartesian meshes to to curvilinear coordinates and complex geometries. The method uses summation by parts with simultaneous approximation terms for computing the diffusion perpendicular to field lines. The diffusion along field lines is computed using a penalty approach, similar to a simultaneous approximation term, but applied across the volume. To extend the method to complex geometry we use a multi-block approach with piecewise smooth structured meshes. That is, the domain is split into sub-grids, with locally adjacent boundaries coupled weakly using penalties. We prove the semi-discrete stability for the curvilinear implementation by deriving discrete energy estimates. The approach is verified though a number of numerical tests, which demonstrate the convergence properties of the method in multi-domain approach. Finally, we present a qualitative result generated in complex geometry and magnetic field, which is generated by the Stepped Pressure Equilibrium Code.