An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics

We introduce an unsplit staggered mesh scheme (USM) that solves multidimensional magnetohydrodynamics (MHD) by a constrained transport method with high-order Godunov fluxes, incorporating three new developments that enhance performance. The USM scheme handles multidimensional MHD terms, proportional to ∇ ·B, in a new directionally unsplit data reconstruction step. This reconstruction step maintains in-plane dynamics very well, as shown by two-dimensional tests. The scheme uses a compact form of the discrete induction equation and since 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 previous electric field averaging schemes and present a new modified electric field construction (MEC) that includes multidimensional derivative information and is more accurate. We also obtain a relation between the induction equation and its difference form and use this to derive a set of corresponding modified equations which show anti-dissipative behavior of cell-face magnetic fields. We present an efficient treatment suppressing the anti-dissipative terms by introducing a difference formulation with balancing dissipation control (DC) that maintains the divergence-free property on a staggered meshes. We use this difference scheme to highlight important properties that avoid unphysical growth of field variables. Our numerical tests show that numerical instability can occur if the anti-dissipation terms are ignored or otherwise not explicitly controlled. A series of comparison studies demonstrates the excellent performance of the full USM-MEC-DC scheme for many quite stringent multidimensional MHD test problems. The scheme is implemented and currently available in the University of Chicago ASC FLASH Center’s FLASH 3 release.