Convergence analysis of operator splitting methods for Maxwell’s equations in dispersive media of Debye type

Abstract

In this paper, two new effective operator splitting methods (SS-MD and SM-MD) for the Maxwell’s equations for dispersive media in two dimensions transverse electric polarization (the 2D Maxwell–Debye TE model) are presented and analyzed. The splitting schemes consist of two sub-stages in each time step, each of which requires solving a number of 1D discrete sub-problems. The Crank–Nicolson approach is used to solve each sub-problem’s time discretization. Both splitting methods satisfy the energy decay and are unconditionally stable. The convergence result of the SS-MD scheme is shown to be of first order in time and of second order in space based on the energy technique, whereas the SM-MD scheme is of second order in both time and space. We also analyze numerical dispersion analysis to obtain two identities of the discrete numerical dispersion relations of both splitting schemes. Examples and numerical experiments are provided to demonstrate and support our theoretical results.