Applications of a finite-volume algorithm for incompressible MHD problems

TL;DR

This paper introduces a stable, parallel finite-volume algorithm for incompressible MHD problems on unstructured grids, validated across various geometries and suitable for benchmarking dynamo simulations.

Contribution

It presents a novel stable mixed Adams-Bashforth/Crank-Nicolson scheme and a pseudo-pressure method for solenoidal magnetic fields in complex geometries.

Findings

Validated accuracy against analytical solutions and previous results.

Implemented boundary conditions for various geometries including spheres and ellipsoids.

Proposed a new simple drifting thermal convection solution for spherical shells.

Abstract

We present the theory, algorithms and implementation of a parallel finite-volume algorithm for the solution of the incompressible magnetohydrodynamic (MHD) equations using unstructured grids that are applicable for a wide variety of geometries. Our method implements a mixed Adams-Bashforth/Crank-Nicolson scheme for the nonlinear terms in the MHD equations and we prove that it is stable independent of the time step. To ensure that the solenoidal condition is met for the magnetic field, we use a method whereby a pseudo-pressure is introduced into the induction equation; since we are concerned with incompressible flows, the resulting Poisson equation for the pseudo-pressure is solved alongside the equivalent Poisson problem for the velocity field. We validate our code in a variety of geometries including periodic boxes, spheres, spherical shells, spheroids and ellipsoids; for the finite geometries we implement the so-called ferromagnetic or pseudo-vacuum boundary conditions appropriate for a surrounding medium with infinite magnetic permeability. This implies that the magnetic field must be purely perpendicular to the boundary. We present a number of comparisons against previous results and against analytical solutions, which verify the code's accuracy. This documents the code's reliability as a prelude to its use in more difficult problems. We finally present a new simple drifting solution for thermal convection in a spherical shell that successfully sustains a magnetic field of simple geometry. By dint of its rapid stabilization from the given initial conditions, we deem it suitable as a benchmark against which other self-consistent dynamo codes can be tested.