A novel approach of divergence-free reconstruction for adaptive mesh refinement

A simple novel approach to maintain the divergence-free condition with adaptive mesh refinement is presented. This new approach uses reconstructions on the coarse faces together with the divergence-free condition to reconstruct the field values on the internal fine faces. It does not construct a global interpolation polynomial over a whole coarse cell. Therefore, it can be easily applied to any refinement ratio. It is implemented via a directionally splitting approach so that it can be applied to any kind of grids and in any dimensions. Implementation is presented in the Cartesian, cylindrical and spherical geometries. It is shown by several 2D magneto-hydrodynamic simulations that such a method can keep the divergence-free error of magnetic fields at the round-off level.