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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, L. Río-Martín, F. Hindenlang, E. Sonnendrücker, M. Dumbser

TL;DR

This work introduces a novel well-balanced, exactly divergence-free semi-implicit hybrid finite volume/finite element scheme for the incompressible viscous and resistive MHD equations on general unstructured mixed-element meshes in 2D and 3D. The method splits the governing equations into convective/viscous and Faraday subsystems, evolving the magnetic field with a divergence-free discrete Stokes formulation while solving the pressure via a vertex-based Poisson problem, all on staggered primal/dual grids. A constrained $L^2$ projection enforces $ abla\cdot\mathbf{B}=0$ for high-order magnetic-field reconstruction, and the equilibrium-based well-balancing ensures exact preservation of prescribed stationary states, enabling accurate long-time simulations in complex tokamak-like geometries. The approach is validated through extensive 2D and 3D tests (Taylor-Green, Euler vortex, MHD vortex) showing second-order accuracy, exact divergence-free magnetic fields, and robust performance on unstructured mixed-element meshes, including non-field-aligned grids. The framework also provides GLM-divergence cleaning as an accessible alternative, and demonstrates applicability to challenging MCF-like configurations with potential impact on plasma physics simulations and fusion research.

Abstract

We present a new divergence-free and well-balanced hybrid FV/FE scheme for the incompressible viscous and resistive MHD equations on unstructured mixed-element meshes in 2 and 3 space dimensions. The equations are split into subsystems. 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 are solved at the aid of an explicit FV scheme, while the magnetic field is evolved in a divergence-free manner via an explicit FV method based on a discrete form of the Stokes law in the edges/faces of each primary element. To achieve higher order of accuracy, a pw-linear polynomial is reconstructed for the magnetic field, which is guaranteed to be divergence-free via a constrained L2 projection. The pressure subsystem is solved implicitly at the aid of a classical continuous FE method in the vertices of the primary mesh. 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. This paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field lines.

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

TL;DR

This work introduces a novel well-balanced, exactly divergence-free semi-implicit hybrid finite volume/finite element scheme for the incompressible viscous and resistive MHD equations on general unstructured mixed-element meshes in 2D and 3D. The method splits the governing equations into convective/viscous and Faraday subsystems, evolving the magnetic field with a divergence-free discrete Stokes formulation while solving the pressure via a vertex-based Poisson problem, all on staggered primal/dual grids. A constrained projection enforces for high-order magnetic-field reconstruction, and the equilibrium-based well-balancing ensures exact preservation of prescribed stationary states, enabling accurate long-time simulations in complex tokamak-like geometries. The approach is validated through extensive 2D and 3D tests (Taylor-Green, Euler vortex, MHD vortex) showing second-order accuracy, exact divergence-free magnetic fields, and robust performance on unstructured mixed-element meshes, including non-field-aligned grids. The framework also provides GLM-divergence cleaning as an accessible alternative, and demonstrates applicability to challenging MCF-like configurations with potential impact on plasma physics simulations and fusion research.

Abstract

We present a new divergence-free and well-balanced hybrid FV/FE scheme for the incompressible viscous and resistive MHD equations on unstructured mixed-element meshes in 2 and 3 space dimensions. The equations are split into subsystems. 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 are solved at the aid of an explicit FV scheme, while the magnetic field is evolved in a divergence-free manner via an explicit FV method based on a discrete form of the Stokes law in the edges/faces of each primary element. To achieve higher order of accuracy, a pw-linear polynomial is reconstructed for the magnetic field, which is guaranteed to be divergence-free via a constrained L2 projection. The pressure subsystem is solved implicitly at the aid of a classical continuous FE method in the vertices of the primary mesh. 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. This paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field lines.
Paper Structure (24 sections, 72 equations, 7 figures, 8 tables)

This paper contains 24 sections, 72 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Governing partial differential equations
  3. Well-balanced numerical method
  4. Convective and viscous subsystem; auxiliary conservative evolution system of the magnetic field
  5. Faraday subsystem
  6. Pressure subsystem
  7. Staggered mixed-element unstructured meshes
  8. Overview of the overall algorithm
  9. Transport-diffusion stage on the primal mesh
  10. Well-balanced reconstruction
  11. Divergence-free evolution of the magnetic field and constrained $L^2$ projection
  12. Divergence-free evolution of the normal components of the magnetic field
  13. Computation of the electric field
  14. Constrained $L^2$ projection
  15. Well-balancing
  16. ...and 9 more sections

Figures (7)

  • Figure 1: Construction of a dual mixed mesh. Left: primal mesh made of two triangles, $T_1$, $T_2$, and two quadrilaterals, $T_3$, $T_4$. The vertex are identified with $\mathtt{V}_k$ while the edges are denoted by $\Gamma_f$. Centre: the barycenters of primal elements are obtained and used to define the triangular subelements related to the faces. Right: merging of the two subelements related to each face results on the corresponding dual element, $C_i$. Interior cells are denoted by $C_2$, $C_3$, $C_5$, $C_8$. Boundary dual cells, $C_1$, $C_4$, $C_6$, $C_7$, $C_9$, $C_{10}$, are constructed to be equal to the boundary subelement.
  • Figure 2: Notation of a tridimensional primal element $T_{1}$: the vertex are identified with $\mathtt{V}_k$ (left), the edges are denoted by $\Lambda_e$ (centre) and the faces are denoted by $\Gamma_f$ (right).
  • Figure 3: Construction of a dual element in 3D based on a face belonging to a hexahedron and a square pyramid. Top left: primal hexahedron (left), primal square pyramid (right), and primal vertex (black dots). Top centre: construction of the barycenters of the primal elements. Top right: each barycenter is connected to the vertex of the common face (shadowed in grey), generating a pyramid on each side of the face with basis the grey face and opposite vertex the barycenters. Bottom left: the lateral faces of the generated pyramids are shaded in light green (left, inside the hexahedron) and sea green (right, inside the primal pyramid). Bottom centre: the vertex of the two new pyramids are marked with green dots; continuous and discontinuous green lines indicate visible and shadow edges when merging both pyramids. Bottom right: the new dual element corresponds to the volume generated by merging the two pyramids constructed.
  • Figure 4: Construction of a dual element in 3D based on a face belonging to a hexahedron and a triangular prism. Top left: primal hexahedron (left), primal triangular prism (right), and primal vertex (black dots). Top centre: construction of the barycenters of the primal elements. Top right: each barycenter is connected to the vertex of the common face (shadowed in grey), generating a pyramid on each side of the face with basis the grey face and opposite vertex the barycenters. Bottom left: the lateral faces of the generated pyramids are shadowed in light green (left, inside the hexahedron) and sea green (right, inside the primal pyramid). Bottom centre: the vertex of the two new pyramids are marked with green dots; continuous and discontinuous green lines indicate visible and shadow edges when merging both pyramids. Bottom right: the new dual element corresponds to the volume generated by merging the two pyramids constructed.
  • Figure 5: Construction of a dual element in 3D based on a face belonging to a square pyramid and a tetrahedron. Top left: primal square pyramid (left), primal tetrahedron (right), and primal vertex (black dots). Top centre: construction of the barycenters of the primal elements. Top right: each barycenter is connected to the vertex of the common face (shaded in grey), generating a tetrahedron on each side of the face with basis the grey face and opposite vertex the barycenters. Bottom left: the lateral faces of the generated tetrahedra are shadowed in light green (left, inside the hexahedron) and sea green (right, inside the primal pyramid). Bottom centre: the vertex of the two new tetrahedra are marked with green dots; continuous and discontinuous green lines indicate visible and shadow edges when merging both tetrahedra. Bottom right: the new dual element corresponds to the volume generated by merging the two tetrahedra constructed.
  • ...and 2 more figures