A componentwise spherical Dirichlet-to-Neumann closure for finite-volume MHD at finite magnetic Reynolds number

Abstract

At finite magnetic Reynolds number, the field induced by a moving conducting fluid or solid extends into open space, so a finite mesh needs an outer boundary condition that represents the decaying exterior field. Magnetohydrodynamic (MHD) codes widely use pseudo-vacuum or truncation at a finite radius. Neither condition is physically correct for a vacuum exterior, but the exact alternatives are costly or limited to simple geometries. We present a Dirichlet-to-Neumann (DtN) closure on a fixed enclosing sphere and release it as part of the open-source multi-region finite-volume solver FreeMHD. The closure matches each Cartesian component of the vector potential to its decaying exterior solution, which retains all three decaying vector-Laplace families. A meshed vacuum buffer connects non-spherical conductors to the sphere. We verify the closure with manufactured modes, sphere free decay, sphere and oblate dynamos, and a freely braking rotor. Under refinement, oblate dynamo thresholds and sphere growth rates converge toward independent spectral references, and the rotor torque matches an exact transient reference. On the same sphere, pseudo-vacuum and truncation give the larger errors predicted by their exterior models.