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Variational discretizations of viscous and resistive magnetohydrodynamics using structure-preserving finite elements

Valentin Carlier

TL;DR

This work develops a structure-preserving, variational discretization for viscous and resistive magnetohydrodynamics (VRMHD) by combining a generalized Lagrange-d'Alembert principle with a metriplectic reformulation. Spatial discretization uses finite element exterior calculus (FEEC) spaces to preserve invariants such as total mass, energy, and the solenoidal constraint $\nabla\cdot\mathbf{B}=0$, while a discrete metriplectic formulation guides a bracket-splitting time integration (Strang splitting) that decouples conservative and dissipative dynamics. Artificial viscosity and resistivity are adaptively scaled with mesh size to stabilize high-gradient regions without erasing important physics. Numerical tests include dispersion relations, ideal and viscoresistive Orszag–Tang vortices, Kelvin–Helmholtz instability, TAEs in tokamak-like geometries, current sheets, and resistive kink modes, demonstrating accurate wave/shock/instability capture and preservation of key invariants, with planned extensions to axis treatment and scalability for fusion-relevant simulations.

Abstract

We propose a novel structure preserving discretization for viscous and resistive magnetohydrodynamics. We follow the recent line of work on discrete least action principle for fluid and plasma equation, incorporating the recent advances to model dissipative phenomena through a generalized Lagrange-d Alembert constrained variational principle. We prove that our semi-discrete scheme is equivalent to a metriplectic system and use this property to propose a Poisson splitting time integration. The resulting approximation preserves mass, energy and the divergence constraint of the magnetic field. We then show some numerical results obtained with our approach. We first test our scheme on simple academic test to compare the results with established methodologies, and then focus specifically on the simulation of plasma instabilities, with some tests on non Cartesian geometries to validate our discretization in the scope of tokamak instabilities.

Variational discretizations of viscous and resistive magnetohydrodynamics using structure-preserving finite elements

TL;DR

This work develops a structure-preserving, variational discretization for viscous and resistive magnetohydrodynamics (VRMHD) by combining a generalized Lagrange-d'Alembert principle with a metriplectic reformulation. Spatial discretization uses finite element exterior calculus (FEEC) spaces to preserve invariants such as total mass, energy, and the solenoidal constraint , while a discrete metriplectic formulation guides a bracket-splitting time integration (Strang splitting) that decouples conservative and dissipative dynamics. Artificial viscosity and resistivity are adaptively scaled with mesh size to stabilize high-gradient regions without erasing important physics. Numerical tests include dispersion relations, ideal and viscoresistive Orszag–Tang vortices, Kelvin–Helmholtz instability, TAEs in tokamak-like geometries, current sheets, and resistive kink modes, demonstrating accurate wave/shock/instability capture and preservation of key invariants, with planned extensions to axis treatment and scalability for fusion-relevant simulations.

Abstract

We propose a novel structure preserving discretization for viscous and resistive magnetohydrodynamics. We follow the recent line of work on discrete least action principle for fluid and plasma equation, incorporating the recent advances to model dissipative phenomena through a generalized Lagrange-d Alembert constrained variational principle. We prove that our semi-discrete scheme is equivalent to a metriplectic system and use this property to propose a Poisson splitting time integration. The resulting approximation preserves mass, energy and the divergence constraint of the magnetic field. We then show some numerical results obtained with our approach. We first test our scheme on simple academic test to compare the results with established methodologies, and then focus specifically on the simulation of plasma instabilities, with some tests on non Cartesian geometries to validate our discretization in the scope of tokamak instabilities.

Paper Structure

This paper contains 24 sections, 6 theorems, 87 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Variational Formulation for Viscous and Resistive Magnetohydrodynamics
  3. Viscous and resistive Magnetohydrodynamics
  4. Variational formulation
  5. Metriplectic reformulation
  6. Discretization
  7. FEEC spaces
  8. Discrete variational principle
  9. Discrete FEM equations
  10. Discrete metriplectic formulation
  11. Bracket splitting and time discretization
  12. Artificial viscosity and resistivity
  13. Numerical Examples
  14. Stabilized ideal tests
  15. Dispersion relation study
  16. ...and 9 more sections

Key Result

Theorem 1

Consider the following Lagrangian : Where $e$ is the internal energy as a function of $\rho$ the density and $s$ the entropy. Solutions to the VRMHD momentum equation correspond to extremal curves of the action Under the variational constraints : with $\boldsymbol{v}$ a time dependent vector field that is null a time $t=0$ and $t=T$ and $[\boldsymbol{u},\boldsymbol{v}] = \boldsymbol{u} \cdot \n

Figures (12)

  • Figure 1: Evolution of the dispersion relation with the use of artificial dissipation
  • Figure 2: Orszag-Tang vortex at $t=2$
  • Figure 3: Orszag-Tang vortex at $t=2$, zoom on the shock with different meshes
  • Figure 4: Density for the Kelvin-Helmholtz instability at $t=2$
  • Figure 5: Domain and mesh for the simulation of the TAE
  • ...and 7 more figures

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Definition 1: Metriplectic system
  • Remark 1
  • Definition 2
  • Theorem 2
  • Remark 2
  • Proposition 3
  • Definition 3
  • Theorem 4
  • ...and 3 more