Referential magnetohydrodynamics: Governing equations

Summary It is shown how the equations of magnetohydrodynamics within an aspherical body diffeomorphic to a ball may be posed with respect to a fixed spherical reference domain. Transforming the equations onto this computational domain produces what we term “referential” equations, which are more algebraically complicated than their physical cousins; on the other hand, the computational gain of this approach is to permit the use of powerful spectral representations of scalar fields on the unit ball, representations that are not readily available in aspherical geometries. The referential equations are derived and schematic discussion is given regarding aspects of their numerical solution. A numerical proof of concept then compares the proposed referential approach with body-fitted spectral-element calculations. For small and moderate departures from spherical symmetry the former is substantially cheaper at fixed accuracy, while its advantage diminishes as the deformation grows. The remaining numerical issues involved in extending this approach to full time-dependent MHD are then discussed.