An Implicit Constrained Transport Method for Magnetic Diffusion in the FLASH Code

Pacific Fusion is using the FLASH Code for target design on our 60 MA pulsed power driver. FLASH is an Eulerian, adaptive mesh refinement (AMR), extended-MHD finite-volume multiphysics code. Simulations of magnetically driven liner implosions with an MHD code, like FLASH, will often model the vacuum dynamics of the magnetic field as a highly resistive medium such that the current density in the vacuum is suitably small compared to the total current in the problem. The parabolic time step restriction, $\Delta t \sim \frac{\eta}{\Delta x ^2}$, associated with the large vacuum resistivity, which is typically much faster than any dynamics of interest, necessitates the use of implicit solvers for the resistive terms in the induction equation. Additionally, multidimensional simulations must also respect the $\nabla\cdot B=0$ constraint for the magnetic field. We present an implementation of a constrained transport method [2] for resistive MHD that evolves magnetic fluxes collocated at cell-faces, which preserves $\nabla\cdot B=0$ to machine precision, including at refinement boundaries. The method solves for the cell-edge integrated electric fields at the $t^{n+1}$ time step by extending the method of Rieben \& White~[1] from finite-elements to a form suitable for use in a finite-volume code. Finally we present benchmark solutions, from both analytic and full physics integrated problems, for our solver implemented in FLASH.