Second-Order-accurate Schemes for Magnetohydrodynamics with Divergence-free Reconstruction

While working on an adaptive mesh refinement (AMR) scheme for divergence-free magnetohydrodynamics (MHD), Balsara discovered a unique strategy for the reconstruction of divergence-free vector fields. Balsara also showed that for one-dimensional variations in flow and field quantities the reconstruction reduces exactly to the total variation diminishing (TVD) reconstruction. In a previous paper by Balsara the innovations were put to use in studying AMR-MHD. While the other consequences of the invention especially as they pertain to numerical scheme design were mentioned, they were not explored in any detail. In this paper we begin such an exploration. We study the problem of divergence-free numerical MHD and show that the work done so far still has four key unresolved issues. We resolve those issues in this paper. It is shown that the magnetic field can be updated in divergence-free fashion with a formulation that is better than the one in Balsara & Spicer. The problem of reconstructing MHD flow variables with spatially second-order accuracy is also studied. Some ideas from weighted essentially non-oscillatory (WENO) reconstruction, as they apply to numerical MHD, are developed. Genuinely multidimensional reconstruction strategies for numerical MHD are also explored. The other goal of this paper is to show that the same well-designed second-order-accurate schemes can be formulated for more complex geometries such as cylindrical and spherical geometry. Being able to do divergence-free reconstruction in those geometries also resolves the problem of doing AMR in those geometries; the appendices contain detailed formulae for the same. The resulting MHD scheme has been implemented in Balsara's RIEMANN framework for parallel, self-adaptive computational astrophysics. The present work also shows that divergence-free reconstruction and the divergence-free time update can be done for numerical MHD on unstructured meshes. As a result, we establish important analogies between MHD on structured meshes and MHD on unstructured meshes because such analogies can guide the design of MHD schemes and AMR-MHD techniques on unstructured meshes. The present paper also lays out the roadmap for designing MHD schemes for structured and unstructured meshes that have better than second-order accuracy in space and time. All the schemes designed here are shown to be second-order-accurate. We also show that the accuracy does not depend on the quality of the Riemann solver. We have compared the numerical dissipation of the unsplit MHD schemes presented here with the dimensionally split MHD schemes that have been used in the past and found the former to be superior. The dissipation does depend on the Riemann solver, but the dependence becomes weaker as the quality of the interpolation is improved. Several stringent test problems are presented to show that the methods work, including problems involving high-velocity flows in low-plasma-β magnetospheric environments. Similar advances can be made in other fields, such as electromagnetics, radiation MHD, and incompressible flow, that rely on a Stokes-law type of update strategy.