ADI schemes for the time integration of Maxwell equations

Abstract

This thesis is concerned with the analysis and construction of alternating direction implicit (ADI) splitting schemes for the time integration of linear isotropic Maxwell equations on cuboids. The work is organized in two major parts. The first part deals with time discrete approximations to exponentially stable Maxwell equations. By means of a divergence cleaning technique and artificial damping, we obtain an ADI scheme with approximations that also decay exponentially in time. The decay rate is here uniform with respect to the time discretization. One of the main ingredients in the proof of the decay behavior is an observability estimate for the numerical approximations. This inequality is obtained by means of a discrete multiplier technique. We also provide a rigorous error analysis for the uniformly exponentially stable ADI scheme, yielding convergence of order one in a space similar to $H^{-1}$. The error result makes only assumptions on the initial data and the model parameters. In the second part, we analyze time discrete approximations to linear isotropic Maxwell equations on a heterogeneous cuboid. In this setting, the domain consists of two different homogeneous subcuboids. The Maxwell equations are here integrated in time by means of the Peaceman-Rachford ADI splitting scheme which is well-known in literature. The main result provides a rigorous error bound of order 3/2 in $L^2$ for the numerical approximations. It is significant that the final error statement involves conditions only on the initial data and model parameters, but not on the solution. To achieve this result, we establish a detailed regularity analysis for the considered Maxwell system.