Constrained-Transport Magnetohydrodynamics with Adaptive-Mesh-Refinement in CHARM

Abstract

We present the implementation of a three-dimensional, second order accurate Godunov-type algorithm for magneto-hydrodynamic (MHD), in the adaptive-mesh-refinement (AMR) cosmological code {\tt CHARM}. The algorithm is based on the full 12-solve spatially unsplit Corner-Transport-Upwind (CTU) scheme. The fluid quantities are cell-centered and are updated using the Piecewise-Parabolic-Method (PPM), while the magnetic field variables are face-centered and are evolved through application of the Stokes theorem on cell edges via a Constrained-Transport (CT) method. The multidimensional MHD source terms required in the predictor step for high-order accuracy are applied in a simplified form which reduces their complexity in three dimensions without loss of accuracy or robustness. The algorithm is implemented on an AMR framework which requires specific synchronization steps across refinement levels. These include face-centered restriction and prolongation operations and a {\it reflux-curl} operation, which maintains a solenoidal magnetic field across refinement boundaries. The code is tested against a large suite of test problems, including convergence tests in smooth flows, shock-tube tests, classical two- and three-dimensional MHD tests, a three-dimensional shock-cloud interaction problem and the formation of a cluster of galaxies in a fully cosmological context. The magnetic field divergence is shown to remain negligible throughout.