Upwind discontinuous Galerkin space discretization and locally implicit time integration for linear Maxwell’s equations

This paper is dedicated to the full discretization of linear Maxwell's equations, where the space discretization is carried out with a discontinuous Galerkin (dG) method on a locally refined spatial grid. For such problems explicit time integrators are ineficient due to their strict CFL condition stemming from the fine grid elements. In the last years this issue of so-called grid-induced stifiness was successfully tackled with locally implicit time integrators. So far, these methods were limited to unstabilized (central uxes) dG methods. However, stabilized (upwind uxes) dG schemes provide many benefits and thus are a popular choice in applications. In this paper we construct a new variant of a locally implicit time integrator using an upwind uxes dG discretization on the coarse part of the grid. The construction is based on a rigorous error analysis which shows that the stabilization operators have to be split differently than the Maxwell operator. Moreover, our earlier analysis of a central uxes locally implicit method based on semigroup theory applies but does not yield optimal convergence rates. In this paper we rigorously prove the stability and provide error bounds of the new method with optimal rates in space and time by means of an energy technique for a suitably defined modified error.