A constrained-transport embedded boundary method for compressible resistive magnetohydrodynamics

TL;DR

The authors develop a constrained-transport embedded-boundary method for compressible resistive MHD on Cartesian grids, enabling arbitrarily shaped, moving, and multi-material interfaces. The induction equation is solved with a staggered constrained-transport scheme, while the hydrodynamic and multi-material coupling use cut-cell and ghost-fluid techniques, with level-set or polyline boundary representations. The method achieves second-order accuracy on smooth problems and first-order accuracy across material discontinuities, and is demonstrated across a broad set of hydrodynamic and MHD test problems, including moving boundaries, two-material interfaces, and various pinch configurations. This approach provides a robust framework for simulating magnetized plasmas in devices with complex geometries, preserving flux and divergence-free magnetic fields in the active domain while handling sharp material interfaces. The work highlights both the practical successes and the current limitations (e.g., numerical resistivity at interfaces and challenges in strongly magnetized, low-Mach regimes) and points toward future extensions to 3D and all-Mach-number formulations.

Abstract

Motivated by the increased interest in pulsed-power magneto-inertial fusion devices in recent years, we present a method for implementing an arbitrarily shaped embedded boundary on a Cartesian mesh while solving the equations of compressible resistive magnetohydrodynamics. The method is built around a finite volume formulation of the equations in which a Riemann solver is used to compute fluxes on the faces between grid cells, and a face-centered constrained transport formulation of the induction equation. The small time step problem associated with the cut cells is avoided by always computing fluxes on the faces and edges of the Cartesian mesh. We extend the method to model a moving interface between two materials with different properties using a ghost-fluid approach, and show some preliminary results including shock-wave-driven and magnetically-driven dynamical compressions of magnetohydrostatic equilibria. We present a thorough verification of the method and show that it converges at second order in the absence of discontinuities, and at first order with a discontinuity in material properties.

Figures (15)

• Figure 1: Illustration of the process of dividing the cells in our cut cell algorithm implementation. The cell type is indicated by cell color (see color map). The polyline representation of the embedded boundary is drawn in white, and white stars mark where it intersects a cell face. The outer and inner faces (lines) and centroids (points) of the cut cells are colored magenta and blue, respectively. The cut face centroids are marked with crosses in the corresponding colors. The boundary normal vectors for the discretized piecewise-linear boundary are drawn as blue arrows.
• Figure 2: Standing acoustic wave in a cylinder imposed using immersed boundary technique.
• Figure 3: Standing magnetosonic wave test in a cylinder using immersed boundaries. The simulation result converges to the exact analytic solution at second order. The setup is a perturbed $\theta$ pinch.
• Figure 4: Standing magnetosonic wave test in a cylinder using immersed boundaries. The setup is a perturbed parallel pinch. The gold solution that was used to compute the error was a high resolution simulation.
• Figure 5: Ideal MHD simulation of the dragging of a magnetic field tangential to an immersed boundary with three different immersed boundary shapes. The domain shown is $32\times 32$ cells. The velocity and magnetic fields are initialy mutually perpendicular, with velocity in the $+\hat{y}$ direction and magnetic field in the $+\hat{x}$ direction.