An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics

We introduce an unsplit staggered mesh scheme (USM) for multidimensional magnetohydrodynamics (MHD) that uses a constrained transport (CT) method with high-order Godunov fluxes and incorporates a new data reconstruction–evolution algorithm for second-order MHD interface states. In this new algorithm, the USM scheme includes so-called “multidimensional MHD terms”, proportional to ∇·B∇·B, in a dimensionally-unsplit way in a single update. This data reconstruction–evolution step, extended from the corner transport upwind (CTU) approach of Colella, maintains in-plane dynamics very well, as shown by the advection of a very weak magnetic field loop in 2D. This data reconstruction–evolution algorithm is also of advantage in its consistency and simplicity when extended to 3D. The scheme maintains the ∇·B=0∇·B=0 constraint by solving a set of discrete induction equations using the standard CT approach, where the accuracy of the computed electric field directly influences the quality of the magnetic field solution. We address the lack of proper dissipative behavior in the simple electric field averaging scheme and present a new modified electric field construction (MEC) that includes multidimensional derivative information and enhances solution accuracy. A series of comparison studies demonstrates the excellent performance of the full USM–MEC scheme for many stringent multidimensional MHD test problems chosen from the literature. The scheme is implemented and currently freely available in the University of Chicago ASC FLASH Center’s FLASH3 release.