A structure preserving numerical method for the ideal compressible MHD system

We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical ener...

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Veröffentlicht in:Journal of computational physics Jg. 508; S. 113009
Hauptverfasser: Dao, Tuan Anh, Nazarov, Murtazo, Tomas, Ignacio
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Inc 01.07.2024
Schlagworte:
Beräkningsvetenskap med inriktning mot numerisk analys
Energy-stability
Invariant domain
Involution constraints
MHD
Scientific Computing with specialization in Numerical Analysis
Structure preserving
Vanishing viscosity limit
Involution constraints
Energy-stability
MHD
Vanishing viscosity limit
Structure preserving
Invariant domain
ISSN:0021-9991, 1090-2716, 1090-2716
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Zusammenfassung:We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wavespeed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering second-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well. •We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations.•The method uses the non-divergence formulation of the MHD system.•The method preserves positivity of density, minimum principle of the specific entropy, energy-stability, and involution constraints.•Euler equations and MHD source system are treated separately allowing for significant freedom in the choice of solvers.•CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wavespeed from Euler's system.
ISSN:0021-9991
1090-2716
1090-2716
DOI:10.1016/j.jcp.2024.113009