Variational discretizations of ideal magnetohydrodynamics in smooth regime using finite element exterior calculus

TL;DR

The paper develops a structure-preserving discretization for the ideal MHD equations in smooth regimes by leveraging a discrete Euler–Poincaré principle and Finite Element Exterior Calculus (FEEC). Vector fields act on differential forms through Lie derivatives, with a discrete hat map ensuring a strong, transport-based discretization that preserves mass, entropy, energy, and the divergence-free condition on the magnetic field. A mid-point time integrator guarantees invariant preservation and reversibility at the fully discrete level. Numerical experiments across barotropic and magnetohydrodynamic test problems demonstrate high-order accuracy and robust invariants maintenance, supporting the method's potential for long-time fusion-plasma simulations. The framework is flexible across FEEC spaces and projection operators, enabling extension to additional physics and scalable implementations.

Abstract

We propose a new class of finite element approximations to ideal compressible magnetohydrodynamic equations in smooth regime. Following variational approximations developed for fluid models in the last decade, our discretizations are built via a discrete variational principle mimicking the continuous Euler-Poincaré principle, and to further exploit the geometrical structure of the problem, vector fields are represented by their action as Lie derivatives on differential forms of any degree. The resulting semi-discrete approximations are shown to conserve the total mass, entropy and energy of the solutions for a wide class of finite element approximations. In addition, the divergence-free nature of the magnetic field is preserved in a pointwise sense and a time discretization is proposed, preserving those invariants and giving a reversible scheme at the fully discrete level. Numerical simulations are conducted to verify the accuracy of our approach and its ability to preserve the invariants for several test problems.

Figures (8)

• Figure 1: Vorticity at $t=0, \ 2, \ 4$, for the double shear layer simulation with a $512 \times 256$ grid, using the discrete spline de Rham sequence \ref{['Discrete_deRham']} with $p=2$, and $\Delta t = 5\mathrm{E{-4}}$.
• Figure 2: Density at $t=0, \ 1, \ 2$, for the fully compressible double shear layer simulation with a $512 \times 256$ grid, using the discrete spline de Rham sequence \ref{['Discrete_deRham']} with degree $p=1$, and $\Delta t = 2\mathrm{E{-4}}$.
• Figure 3: Error in density and velocity after reversing the scheme at $t=2$ for the fully compressible double shear layer simulation, using the spline sequence \ref{['Discrete_deRham']} with degree $p=1$ on a $256 \times 128$ grid, and $\Delta t = 2\mathrm{E{-4}}$.
• Figure 4: Evolution of the density for the Rayleigh-Taylor instability, for the times $t=2.0,\ 2.4,\ 2.8,\ 3.2,\ 3.6,\ 4.0$. This dissipation-less simulation uses the spline spaces \ref{['Discrete_deRham']} with $p=1$ on a $128 \times 512$ grid, and $\Delta t = 2\mathrm{E{-4}}$.
• Figure 5: Plot of $v_z$, $B_z$, $v_{\perp} = \cos \alpha v_y -\sin \alpha v_x$ and $B_{\perp} = \cos \alpha B_y -\sin \alpha B_x$ for the Alfvén wave at $t=1$ (after one period of the wave), cuts at $y=0$. The squares are the reference (discretization with $N=64$ cells at $t=0$) the crosses are a coarse discretization with $N=16$ cells at $t=1$ and the dash line is a finer discretization with $N=64$ cells also at $t=1$.

Theorems & Definitions (26)

• Theorem 1
• Remark 1
• proof
• Definition 1: General discrete variational problem
• Definition 2: Restricted discrete variational problem
• Remark 2
• Proposition 2
• proof
• Proposition 3: semi-discrete scheme
• proof