Asymptotic-preserving semi-implicit finite volume scheme for Extended Magnetohydrodynamics

TL;DR

This work develops an asymptotic-preserving, semi-implicit finite-volume solver for Extended MHD (XMHD) that preserves the ideal MHD structure while incorporating Hall drift and electron inertia. By reformulating XMHD so that the ideal MHD fluxes are maintained on the left-hand side and the extended physics appear as implicit source terms, the method reuses standard ideal MHD solvers and constrained transport to maintain $ abladot extbf{B}=0$. A density-dependent slope limiter, along with directional splitting of implicit updates and edge-centered variable placement, stabilizes the scheme across low- and high-density regimes and enables scalable AMR performance. The solver is validated against Brio–Wu shock-tube, ideal MHD linear waves, resistive diffusion, Hall-drift problems, and 2D nonlinear tests, demonstrating correct limiting behavior, second-order convergence in ideal and resistive limits, and accurate Hall-dominated dynamics. This approach enables robust XMHD simulations across wide density and regime variations in high-energy-density plasmas, with opportunities for further accuracy and stability enhancements in future work.

Abstract

A Finite Volume (FV) scheme is developed for solving the extended magnetohydrodynamic (XMHD) equations, yielding accurate results in the ideal, resistive, and Hall MHD limits. This is accomplished by first re-writing the XMHD equations such that it allows the algorithm to retain the use of ideal MHD Riemann solvers and the constrained transport method to preserve divergence-free magnetic fields. Incorporation of electron inertia and displacement current introduces additional numerical stiffness which motivates a semi-implicit FV scheme that re-formulates the XMHD model as a relaxation system. The equations are then advanced in time using an explicit 2nd-order Runge-Kutta scheme with operator splitting applied to the implicit source term updates at each sub-stage. For additional numerical stability, a density-dependent slope limiter is implemented to increase flux diffusivity at low density regions where non-ideal effects become significant. The algorithm is subsequently implemented in a scalable adaptive mesh refinement (AMR) framework. As the new algorithm retains many aspects of the ideal MHD formulations, it asymptotes naturally to the ideal MHD limit. Moreover, it shows promising results at the resistive and Hall MHD limits. This is verified against reference test problems for ideal, resistive and Hall MHD.

Figures (15)

• Figure 1: A 3D computational cell on a Cartesian grid is illustrated to indicate the positions of various field variables. The cell center is marked with an ‘x’, while the arrows normal to each cell face represent F1, F2, and F3, corresponding to faces with normal directions along the x, y, and z axes, respectively. The cell edges are shown by arrows labeled E1, E2, and E3, indicating edges oriented along the x, y, and z directions. The cell nodes are depicted as solid black dots and are denoted by NN. $\mathbf{B}$ is placed on the faces (F) while both $\mathbf{E}$ and $\mathbf{J}$ lie on the edges (E). The rest of the conserved variables are placed at the cell-center ($i,j,k$).
• Figure 2: 2D diagram illustrating the faces $F3$ of six computational cells, with their normals directed into the paper along the z-axis. In this configuration, the faces $F3(i,j,k-\frac{1}{2})$ and $F3(i,j-1,k-\frac{1}{2})$ correspond to cell centers at $(i,j,k)$ and $(i,j-1,k)$, respectively. Each computational cell is bounded by edges that define its geometry: the horizontal arrows represent the edges $E1$, while the vertical arrows denote the edges $E2$. The edges $E3$, which align with the z-axis, are positioned at the cell nodes and point into the paper. The intersections of these edges are depicted as solid black dots to indicate cell nodes $NN$.
• Figure 3: Brio-Wu shock tube solution from Zhao et. al. ZHAO2014400 using LLF Riemann solver at cell interface whereby the left figure (a) contains oscillation due to whistler wave introduced by Hall effect which is important at low density. However, the figure (c) on the right is unable to replicate the same solution as figure (b) at the ideal MHD limit. Figures (d) and (e) are produced using XMHD (LLF) on Artemis and compared against Athena++ ideal MHD solver (LLF) to show the increment of non-ideal effects at lower density. Different CFL numbers are also used to ensure that the solution is consistent at different time resolution.
• Figure 4: For the standard ideal MHD test case on shock tube, the XMHD (Artemis) is quite similar to ideal MHD (Athena++) but there is some discrepancy especially at lower-density region where non-ideal effects become more crucial.
• Figure 5: At low density, XMHD (Artemis) begins to deviate significantly from ideal MHD (Athena++) due to more significant non-ideal MHD effects.