Efficient time integration of the Maxwell–Klein–Gordon equation in the non-relativistic limit regime

摘要

The Maxwell–Klein–Gordon equation describes the interaction of a charged particle with an electromagnetic field. Solving this equation in the non-relativistic limit regime, i.e. the speed of light c formally tending to infinity, is numerically very delicate as the solution becomes highly-oscillatory in time. In order to resolve the oscillations, standard numerical time integration schemes require severe time step restrictions depending on the large parameter c2.The idea to overcome this numerical challenge is to filter out the high frequencies explicitly by asymptotically expanding the exact solution with respect to the small parameter c−2. This allows us to reduce the highly-oscillatory problem to its corresponding non-oscillatory Schrödinger–Poisson limit system. On the basis of this expansion we are then able to construct efficient numerical time integration schemes, which do NOT suffer from any c-dependent time step restriction.