2024/07/19 by Francesco Fambri, Éric Sonnendrücker, Fambri, Francesco +1
Engineering · Mathematics · #65M08 #65M60 #76M10 #76M12 (Secondary) #85-08 (Primary) 65M12 #Computational Fluid Dynamics and Aerodynamics #F.2.1 #FOS: Mathematics #FOS: Physical sciences #G.1 #Gas Dynamics and Kinetic Theory #Instrumentation and Methods for Astrophysics (astro-ph.IM) #J.2 #Numerical Analysis (math.NA) #Plasma Physics (physics.plasm-ph) #Plasma and Flow Control in Aerodynamics
paper · pdf · doi:10.48550/arxiv.2407.14141
openalex publication_date 2024/07/19 · openalex created_date 2024/09/26 · openalex updated_date 2026/07/28
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 Alfvénic 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 Alfvénic 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.