Relativistic magnetohydrodynamics in dynamical spacetimes: A new adaptive mesh refinement implementation

We have written and tested a new general relativistic magnetohydrodynamics code, capable of evolving magnetohydrodynamics (MHD) fluids in dynamical spacetimes with adaptive-mesh refinement (AMR). Our code solves the Einstein-Maxwell-MHD system of coupled equations in full $3+1$ dimensions, evolving the metric via the Baumgarte-Shapiro Shibata-Nakamura formalism and the MHD and magnetic induction equations via a conservative, high-resolution shock-capturing scheme. The induction equations are recast as an evolution equation for the magnetic vector potential, which exists on a grid that is staggered with respect to the hydrodynamic and metric variables. The divergenceless constraint $\mathbf{\ensuremath{\nabla}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{B}=0$ is enforced by the curl of the vector potential. Our MHD scheme is fully compatible with AMR, so that fluids at AMR refinement boundaries maintain $\mathbf{\ensuremath{\nabla}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{B}=0$. In simulations with uniform grid spacing, our MHD scheme is numerically equivalent to a commonly used, staggered-mesh constrained-transport scheme. We present code validation test results, both in Minkowski and curved spacetimes. They include magnetized shocks, nonlinear Alfv\'en waves, cylindrical explosions, cylindrical rotating disks, magnetized Bondi tests, and the collapse of a magnetized rotating star. Some of the more stringent tests involve black holes. We find good agreement between analytic and numerical solutions in these tests, and achieve convergence at the expected order.