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Multi-dimensional upwind constrained transport on unstructured grids for 'shallow water' magnetohydrodynamics

Novel Multi-dimensional Upwind Constrained Transport (MUCT) schemes on un-structured triangular grids are described. Constrained Transport (CT) discretizations conserve the divergence-free nature of divergence-free vector fields on the discrete level. Multi-dimensional Upwind (MU) schemes generalize the concept of dimensionally split upwind schemes for hyperbolic systems to truly multidimensional upwind discretizations on unstructured grids with compact stencils consisting of nearest neighbors. In the present paper the concept of Constrained Transport, generalized to unstructured triangular grids using face elements, is combined with the concept of Multi-dimensional Upwind advection schemes. The resulting MUCT schemes are applied to flow simulations on 2D triangular unstructured grids. The schemes are applied to the numerical solution of Faraday's law of induction in the Magnetohydropdynamic (MHD) approximation , which describes the dynamical evolution of a divergence-free magnetic field, and to the numerical solution of the the recently proposed hyperbolic system formed by the 'Shal-low Water' Magnetohydrodynamics equations. It is described how the two-dimensional MUCT schemes presented can be generalized to flow simulation on three-dimensional tetrahedral grids. The MUCT schemes can be applied to any hyperbolic system with divergence-free fields, including the full MHD equations. In the MUCT schemes the CT approach is generalized to multi-dimensional methods on unstructured grids.