Temporal Substepping Scheme for Magnetohydrodynamics with Cell-based Adaptive Mesh Refinement on Staggered Grid

We present a new algorithm for numerical magnetohydrodynamics on staggered meshes preserving $\mathrm{\nabla }\cdot B=0$. Our algorithm is based on the constrained transport method and supports both cell-based adaptive mesh refinement and temporal substepping. We handle resolution changes directly on the logically Cartesian grid without needing interpolation or projection between nested or neighboring grids, nor coupling the solution between refinement levels.