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