Abstract.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.

This paper is dedicated to the improvement of the efficiency of the leap-frog method for second order differential equations. In numerous situations the strict CFL condition of the leapfrog method is the main bottleneck that thwarts its performance. Based on Chebyshev polynomials new methods have been constructed that exhibit a much weaker CFL condition than the leap-frog method. However, these methods do not even approximately conserve the energy of the exact solution which can result in a bad approximation quality.In this paper we propose two remedies to this drawback. For linear problems we show by using energy techniques that damping the Chebyshev polynomial leads to approximations which approximately preserve a discrete energy norm over arbitrary long times. Moreover, with a completely different approach based on generating functions, we propose to use special starting values that considerably improve the stability. We show that the new schemes arising from these modifications are of order two and can be modified to be of order four. These convergence results apply to semilinear problems. Finally, we discuss the efficient implementation of the new schemes and give generalizations to fully nonlinear equations.

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