Locally Implicit Time Integration for Linear Maxwell's Equations

Abstract

This thesis is concerned with the full discretization of Maxwell's equations in cases where the spatial discretization has to be carried out with a locally refined grid. In such situations locally implicit time integrators are an appealing choice for the time discretization since they overcome the grid-induced stiffness of these problems. We analyze such a locally implicit time integrator in the case where the space discretization stems from a central fluxes discontinuous Galerkin method. In fact, we prove its stability under a CFL condition which solely depends on the coarse part of the spatial grid and give a rigorous error analysis showing that the integrator is second order convergent. Moreover, we extend this time integrator so that it can be applied to an upwind fluxes discontinuous Galerkin space discretization. We show that this novel integrator preserves the second order temporal convergence and that it inherits the improved properties of an upwind fluxes discretization (better stability, higher spatial convergence rate) compared to the central fluxes case.