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

We present a new exactly divergence-free and well-balanced hybrid finite volume/finite element scheme for the numerical solution of the incompressible viscous and resistive magnetohydrodynamics (MHD) equations on staggered unstructured mixed-element meshes in two and three space dimensions. The equa...

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Published inJournal of computational physics Vol. 493; p. 112493
Main Authors Fambri, F., Zampa, E., Busto, S., Río-Martín, L., Hindenlang, F., Sonnendrücker, E., Dumbser, M.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.11.2023
Subjects
Divergence-free
Incompressible viscous and resistive magnetohydrodynamics
Semi-implicit hybrid finite volume / finite element scheme
Staggered unstructured mixed-element meshes
Well-balanced
Semi-implicit hybrid finite volume / finite element scheme
Divergence-free
Incompressible viscous and resistive magnetohydrodynamics
Staggered unstructured mixed-element meshes
Well-balanced
Online AccessGet full text
ISSN0021-9991
1090-2716
1090-2716
DOI10.1016/j.jcp.2023.112493

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Abstract We present a new exactly divergence-free and well-balanced hybrid finite volume/finite element scheme for the numerical solution of the incompressible viscous and resistive magnetohydrodynamics (MHD) equations on staggered unstructured mixed-element meshes in two and three space dimensions. The equations are split into several subsystems, each of which is then discretized with a particular scheme that allows to preserve some fundamental structural features of the underlying governing PDE system also at the discrete level. The pressure is defined on the vertices of the primary mesh, while the velocity field and the normal components of the magnetic field are defined on an edge-based/face-based dual mesh in two and three space dimensions, respectively. This allows to account for the divergence-free conditions of the velocity field and of the magnetic field in a rather natural manner. The non-linear convective and the viscous terms in the momentum equation are solved at the aid of an explicit finite volume scheme, while the magnetic field is evolved in an exactly divergence-free manner via an explicit finite volume method based on a discrete form of the Stokes law in the edges/faces of each primary element. The latter method is stabilized by the proper choice of the numerical resistivity in the computation of the electric field in the vertices/edges of the 2D/3D elements. To achieve higher order of accuracy, a piecewise linear polynomial is reconstructed for the magnetic field, which is guaranteed to be exactly divergence-free via a constrained L2 projection. Finally, the pressure subsystem is solved implicitly at the aid of a classical continuous finite element method in the vertices of the primary mesh and making use of the staggered arrangement of the velocity, which is typical for incompressible Navier-Stokes solvers. In order to maintain non-trivial stationary equilibrium solutions of the governing PDE system exactly, which are assumed to be known a priori, each step of the new algorithm takes the known equilibrium solution explicitly into account so that the method becomes exactly well-balanced. We show numerous test cases in two and three space dimensions in order to validate our new method carefully against known exact and numerical reference solutions. In particular, this paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields and the obtained numerical solutions are provided as free supplementary electronic material to allow other research groups to reproduce our results and to compare with our data. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations, showing that the new well-balanced scheme introduced in this paper is able to maintain stationary equilibria exactly over very long integration times even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field. •Semi-implicit FV/FE method for incompressible viscous and resistive MHD equations.•Well-balanced and exactly divergence-free on general unstructured mixed-element grids.•Constrained L2 projection for an exactly divergence-free reconstruction.•Thorough study of the lid-driven MHD cavity problem (reference solution is provided).•Stable long-time simulation of Grad-Shafranov equilibria in 3D tokamak geometries.
AbstractList We present a new exactly divergence-free and well-balanced hybrid finite volume/finite element scheme for the numerical solution of the incompressible viscous and resistive magnetohydrodynamics (MHD) equations on staggered unstructured mixed-element meshes in two and three space dimensions. The equations are split into several subsystems, each of which is then discretized with a particular scheme that allows to preserve some fundamental structural features of the underlying governing PDE system also at the discrete level. The pressure is defined on the vertices of the primary mesh, while the velocity field and the normal components of the magnetic field are defined on an edge-based/face-based dual mesh in two and three space dimensions, respectively. This allows to account for the divergence-free conditions of the velocity field and of the magnetic field in a rather natural manner. The non-linear convective and the viscous terms in the momentum equation are solved at the aid of an explicit finite volume scheme, while the magnetic field is evolved in an exactly divergence-free manner via an explicit finite volume method based on a discrete form of the Stokes law in the edges/faces of each primary element. The latter method is stabilized by the proper choice of the numerical resistivity in the computation of the electric field in the vertices/edges of the 2D/3D elements. To achieve higher order of accuracy, a piecewise linear polynomial is reconstructed for the magnetic field, which is guaranteed to be exactly divergence-free via a constrained L2 projection. Finally, the pressure subsystem is solved implicitly at the aid of a classical continuous finite element method in the vertices of the primary mesh and making use of the staggered arrangement of the velocity, which is typical for incompressible Navier-Stokes solvers. In order to maintain non-trivial stationary equilibrium solutions of the governing PDE system exactly, which are assumed to be known a priori, each step of the new algorithm takes the known equilibrium solution explicitly into account so that the method becomes exactly well-balanced. We show numerous test cases in two and three space dimensions in order to validate our new method carefully against known exact and numerical reference solutions. In particular, this paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields and the obtained numerical solutions are provided as free supplementary electronic material to allow other research groups to reproduce our results and to compare with our data. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations, showing that the new well-balanced scheme introduced in this paper is able to maintain stationary equilibria exactly over very long integration times even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field. •Semi-implicit FV/FE method for incompressible viscous and resistive MHD equations.•Well-balanced and exactly divergence-free on general unstructured mixed-element grids.•Constrained L2 projection for an exactly divergence-free reconstruction.•Thorough study of the lid-driven MHD cavity problem (reference solution is provided).•Stable long-time simulation of Grad-Shafranov equilibria in 3D tokamak geometries.
ArticleNumber 112493
Author Hindenlang, F.
Sonnendrücker, E.
Zampa, E.
Busto, S.
Río-Martín, L.
Dumbser, M.
Fambri, F.
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  surname: Fambri
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  surname: Zampa
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  organization: Department of Mathematics, University of Trento, Via Sommarive 14, 38123 Trento, Italy
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  givenname: S.
  surname: Busto
  fullname: Busto, S.
  email: saray.bustoulloa@unitn.it
  organization: Department of Applied Mathematics I, Universidade de Vigo, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain
– sequence: 4
  givenname: L.
  surname: Río-Martín
  fullname: Río-Martín, L.
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  surname: Hindenlang
  fullname: Hindenlang, F.
  email: florian.hindenlang@ipp.mpg.de
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  surname: Sonnendrücker
  fullname: Sonnendrücker, E.
  email: sonnen@ipp.mpg.de
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– sequence: 7
  givenname: M.
  surname: Dumbser
  fullname: Dumbser, M.
  email: michael.dumbser@unitn.it
  organization: Laboratory of Applied Mathematics, DICAM, University of Trento, via Mesiano 77, 38123 Trento, Italy
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Keywords Semi-implicit hybrid finite volume / finite element scheme
Divergence-free
Incompressible viscous and resistive magnetohydrodynamics
Staggered unstructured mixed-element meshes
Well-balanced
Language English
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Snippet We present a new exactly divergence-free and well-balanced hybrid finite volume/finite element scheme for the numerical solution of the incompressible viscous...
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StartPage 112493
SubjectTerms Divergence-free
Incompressible viscous and resistive magnetohydrodynamics
Semi-implicit hybrid finite volume / finite element scheme
Staggered unstructured mixed-element meshes
Well-balanced
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Title A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume / finite element scheme for the incompressible MHD equations
URI https://dx.doi.org/10.1016/j.jcp.2023.112493
https://doi.org/10.1016/j.jcp.2023.112493
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