A fourth-order adaptive mesh refinement solver for Maxwell's Equations

We present a fourth-order accurate, multilevel Maxwell solver, discretized in space with a finite volume approach and advanced in time with the classical fourth-order Runge Kutta method (RK4). Electric fields are decomposed into divergence-free and curl-free parts; we solve for the divergence-free parts of Faraday's Law and the Amp\\`ere-Maxwell Law while imposing Gauss' Laws as initial conditions. We employ a damping scheme inspired by the Advanced Weather Research and Forecasting Model to eliminate non-physical waves reflected off of coarse-fine grid boundaries, and Kreiss-Oliger artificial dissipation to remove standing wave instabilities. Surprisingly, artificial dissipation appears to damp the spuriously reflected waves at least as effectively as the atmospheric community's damping scheme.