Error analysis of the implicit Euler scheme for the Maxwell–Kerr system

Abstract

We establish first-order convergence of the implicit Euler scheme for the quasilinear Maxwell equations with Kerr-type material laws. We only impose regularity assumption which are in accordance with the newly established wellposed theory for the PDE system. In recent literure CFL conditions had to be imposed on full discretizations of this system even for implicit time integration schemes. In our results on the semi- discretization, the time step size is only restricted by the $\mathcal{H}^3$-norm $r_0$ of the initial fields, and the solutions of the scheme are bounded by $c(r_0)$. We thus expect to obtain full discretization results without CFL condition in future work. The estimates are shown by an intricate iterative procedure inspired by the methods used in the wellposedness theory of the PDE