Divergence-free magnetohydrodynamics on conformally moving, adaptive meshes using a vector potential method

We present a new method for evolving the equations of magnetohydrodynamics (both Newtonian and relativistic) that is capable of maintaining a divergence-free magnetic field ($\nabla \cdot B=0$) on adaptively refined, conformally moving meshes. The method relies on evolving the magnetic vector potential and then using it to reconstruct the magnetic fields. The advantage of this approach is that the vector potential is not subject to a constraint equation in the same way the magnetic field is, and so can be refined and moved in a straightforward way. We test this new method against a wide array of problems from simple Alfvén waves on a uniform grid to general relativistic MHD simulations of black hole accretion on a nested, spherical-polar grid. We find that the code produces accurate results and in all cases maintains a divergence-free magnetic field to machine precision.