A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume / finite element scheme for the incompressible MHD equations

Fambri, F.; Zampa, E.; Busto, S.; Río-Martín, L.; Hindenlang, F.; Sonnendrücker, E.; Dumbser, M.

doi:10.1016/j.jcp.2023.112493

Journal of Computational Physics

2023

DOI: 10.1016/j.jcp.2023.112493

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A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume / finite element scheme for the incompressible MHD equations

F. Fambri,

E. Zampa,

S. Busto

et al.

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2024

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Cited by 7 publications

References 129 publications

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A Well-Balanced Semi-implicit IMEX Finite Volume Scheme for Ideal Magnetohydrodynamics at All Mach Numbers

Birke,

Boscheri,

Klingenberg

2024

J Sci Comput

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We propose a second-order accurate semi-implicit and well-balanced finite volume scheme for the equations of ideal magnetohydrodynamics including gravitational source terms. The scheme treats all terms associated with the acoustic pressure implicitly while keeping the remaining terms part of the explicit sub-system. This semi-implicit approach makes the method particularly well suited for problems in the low Mach regime. We combine the semi-implicit scheme with the deviation well-balancing technique and prove that it maintains equilibrium solutions for the magnetohydrostatic case up to rounding errors. In order to preserve the divergence-free property of the magnetic field enforced by the solenoidal constraint, we incorporate a constrained transport method in the semi-implicit framework. Second order of accuracy is achieved by means of a standard spatial reconstruction technique with total variation diminishing property, and by an asymptotic preserving time stepping algorithm built upon the implicit-explicit Runge–Kutta time integrators. Numerical tests in the low Mach regime and near magnetohydrostatic equilibria support the low Mach and well-balanced properties of the numerical method.

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A Well-Balanced Semi-implicit IMEX Finite Volume Scheme for Ideal Magnetohydrodynamics at All Mach Numbers

Birke,

Boscheri,

Klingenberg

2024

J Sci Comput

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A Structure-Preserving Semi-implicit IMEX Finite Volume Scheme for Ideal Magnetohydrodynamics at all Mach and Alfvén Numbers

Boscheri,

Thomann

2024

J Sci Comput

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We present a divergence-free semi-implicit finite volume scheme for the simulation of the ideal magnetohydrodynamics (MHD) equations which is stable for large time steps controlled by the local transport speed at all Mach and Alfvén numbers. An operator splitting technique allows to treat the convective terms explicitly while the hydrodynamic pressure and the magnetic field contributions are integrated implicitly, yielding two decoupled linear implicit systems. The linearity of the implicit part is achieved by means of a semi-implicit time linearization. This structure is favorable as second-order accuracy in time can be achieved relying on the class of semi-implicit IMplicit–EXplicit Runge–Kutta (IMEX-RK) methods. In space, implicit cell-centered finite difference operators are designed to discretely preserve the divergence-free property of the magnetic field on three-dimensional Cartesian meshes. The new scheme is also particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, since no explicit numerical dissipation is added to the implicit contribution and the time step is scale independent. Likewise, highly magnetized flows can benefit from the implicit treatment of the magnetic fluxes, hence improving the computational efficiency of the novel method. The convective terms undergo a shock-capturing second order finite volume discretization to guarantee the effectiveness of the proposed method even for high Mach number flows. The new scheme is benchmarked against a series of test cases for the ideal MHD equations addressing different acoustic and Alfvén Mach number regimes where the performance and the stability of the new scheme is assessed.

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A divergence-free hybrid finite volume / finite element scheme for the incompressible MHD equations based on compatible finite element spaces with a posteriori limiting

Zampa,

Busto,

Dumbser

2024

Applied Numerical Mathematics

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A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume / finite element scheme for the incompressible MHD equations

Cited by 7 publications

References 129 publications

A Well-Balanced Semi-implicit IMEX Finite Volume Scheme for Ideal Magnetohydrodynamics at All Mach Numbers

A Well-Balanced Semi-implicit IMEX Finite Volume Scheme for Ideal Magnetohydrodynamics at All Mach Numbers

A Structure-Preserving Semi-implicit IMEX Finite Volume Scheme for Ideal Magnetohydrodynamics at all Mach and Alfvén Numbers

A divergence-free hybrid finite volume / finite element scheme for the incompressible MHD equations based on compatible finite element spaces with a posteriori limiting

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