Locally Implicit Time Integration for Linear Maxwell's Equations

Error analysis of a fully discrete discontinuous Galerkin alternating direction implicit discretization of a class of linear wave-type problems

“…We show that, if the solution is sufficiently regular, these approximations fulfill error bounds of the same order as the approximations to the solution given directly by the scheme. We point out that these results can be generalized in a straightforward way to other schemes that can be analyzed by the techniques from [24,40]. As far as we are aware of, such results were not given before in the literature.…”

“…Such an improved order of convergence on a regular rectangular grid was shown in [30] for the neutron transport equation in dimension two. For general meshes and less regular solutions this might not be the case, see e.g., [40,Fig. 3.3] for a counterexample on unstructured meshes.…”

“…The analysis we present here is based on the fact that the Peaceman-Rachford scheme can be interpreted as a perturbation of the Crank-Nicolson scheme. Hence, we were able to extend techniques established in [24,40]. These techniques were originally developed for a locally implicit time integration scheme but also apply to the full discretization obtained by the Crank-Nicolson and the leapfrog scheme, respectively.…”