Error Analysis of a Second-Order Locally Implicit Method for Linear Maxwell's Equations

In this paper we consider the full discretization of linear Maxwell’s equations on spatial grids which are locally refined. For such problems, explicit time integration schemes become ine cient because the smallest mesh width results in a strict CFL condition. Recently locally implicit time integration methods have become popular in overcoming the problem of so called grid-induced sti↵ness. Various such schemes have been proposed in the literature and have been shown to be very e cient. However, a rigorous analysis of such methods is still missing. In fact, the available literature focuses on error bounds which are valid on a fixed spatial mesh only but deteriorate in the limit where the smallest spatial mesh size tends to zero. Moreover, some important questions cannot be answered without such an analysis. For example, it has not yet been studied which elements of the spatial mesh enter the CFL condition. In this paper we provide such a rigorous analysis for a locally implicit scheme proposed by Verwer [15] based on a variational formulation and energy techniques.