High-Resolution Finite Element Schemes for the Compressible MHD Equations

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The goal of this project is the development and analysis of high-resolution finite element schemes for solving the equations of ideal magnetohydrodynamics (MHD) on unstructured meshes in 3D. The main focus is on the design of algorithms satisfying all problem-specific physical constraints (conservation laws, maximum principles, divergence-free conditions) at the discrete level. The proposed approach belongs to the family of staggered constrained transport (CT) methods. A physics-compatible choice of nodal, edge, and face finite element spaces makes it possible to avoid divergence errors that cause numerical instabilities. A nonoscillatory approximation to the compressible MHD system is constructed using a new type of artificial dissipation and an element-based version of the flux-corrected transport (FCT) algorithm. The continuous nodal approximation of the conserved quantities is used to calculate the edge-based electric field intensity and the face-based magnetic flux density from the Ohm and Maxwell-Faraday laws. The formation of spurious undershoots/overshoots is prevented using a custom-made limiting strategy. The proposed MHD solver can be run in an implicit or explicit mode. It does not require dimensional splitting and produces solenoidal magnetic fields without the cost of solving a Poisson equation or a transport equation for the vector-valued magnetic potential. The developed software will be used to study idealized magnetic Z-pinch liner implosions and 3D magnetic Rayleigh-Taylor (MRT) instabilities.