Structure preserving hybrid Finite Volume Finite Element method for compressible MHD

TL;DR

This paper introduces a novel hybrid finite volume and finite element numerical method for compressible magnetohydrodynamics that is stable, accurate across Mach regimes, and preserves key physical properties at the discrete level.

Contribution

The paper develops a semi-implicit splitting scheme combining FV and FE discretizations using FEEC, ensuring energy stability and divergence-free magnetic fields in all Mach regimes.

Findings

Robust and accurate in low and high Mach regimes

Efficient linear algebraic systems solved by conjugate-gradient method

Preserves energy stability and magnetic-helicity conservation

Abstract

In this manuscript we present a novel and efficient numerical method for the compressible viscous and resistive MHD equations for all Mach number regimes. The time-integration strategy is a semi-implicit splitting, combined with a hybrid finite-volume and finite-element (FE) discretization in space. The non-linear convection is solved by a robust explicit FV scheme, while the magneto-acoustic terms are treated implicitly in time. The resulting CFL stability condition depends only on the fluid velocity, and not on the Alfv\'enic and acoustic modes. The magneto-acoustic terms are discretized by compatible FE based on a continuous and a discrete de Rham complexes designed using Finite Element Exterior Calculus (FEEC). Thanks to the use of FEEC, energy stability, magnetic-helicity conservation and the divergence-free conditions can be preserved also at the discrete level. A very efficient splitting approach is used to separate the acoustic and the Alfv\'enic modes in such a fashion that the original symmetries of the PDE governing equations are preserved. In this way, the algorithm relies on the solution of linear, symmetric and positive-definite algebraic systems, that are very efficiently handled by the simple matrix-free conjugate-gradient method. The resulting algorithm showed to be robust and accurate in low and high Mach regimes even at large Courant numbers. Non-trivial tests are solved in one-, two- and three- space dimensions to confirm the robustness, accuracy, and the low-dissipative and conserving properties of the final algorithm. While the formulation of the method is very general, numerical results for a second-order accurate FV-FE scheme will be presented.