The numerical solution of Maxwell's equations in second-order form is of fundamental importance to the simulation of time dependent electromagnetic phenomena. To adress the wide range of difficulties involved, the model equation is discretized in space by symmetric interior penalty discontinuous Galerkin methods, which yield an essentially diagonal mass matrix.
In the presence of complex geometry, adaptivity and mesh refinement are certainly key for the efficient numerical solution of partial differential equations. However, locally refined meshes impose severe stability constraints on explicit time-stepping schemes, where the maximal time-step allowed by a CFL condition is dictated by the smallest elements in the mesh. When mesh refinement is restricted to a small region, the use of implicit methods, or a very small time step in the entire computational domain, are very high a price to pay. To overcome the stability restriction, we propose a local time-stepping scheme, which allow arbitrarily small time-steps precisely where small elements in the mesh are located. Starting from the standard second-order ``leap-frog'' scheme, a time-stepping method of second-order of accuracy is derived. Numerical experiments illustrate the efficiency of the method and validate the theory.