---
abstract: |
    Based on the Suzuki product-formula approach, we construct a family of unconditionally stable algorithms to solve the time-dependent Maxwell equations. We describe a practical implementation of these algorithms for one-, two-, and three-dimensional systems with spatially varying permittivity and permeability. The salient features of the algorithms are illustrated by computing selected eigenmodes and the full density of states of one-, two-, and three-dimensional models and by simulating the propagation of light in slabs of photonic band-gap materials.

    PACS numbers: 02.60.Cb, 03.50.De, 41.20.Jb, 42.70.Qs
author:
- |
    Sebastiaan Kole, Marc Thilo Figge, and Hans De Raedt\
    Centre for Theoretical Physics and Materials Science Centre\
    University of Groningen, Nijenborgh 4\
    NL-9747 AG Groningen, The Netherlands\
    E-mail: j.s.kole@@phys.rug.nl, m.t.figge@@phys.rug.nl, deraedt@@phys.rug.nl\
    http://rugth30.phys.rug.nl/compphys
title: |
    Unconditionally Stable Algorithms to Solve the\
    Time-Dependent Maxwell Equations
---

Introduction {#sec1}
============

Maxwell’s partial differential equations of electromagnetism describe the evolution of electric and magnetic fields in time and space [@BornWolf]. They apply to a wide range of different physical situations that are specified by the boundary conditions on the electromagnetic (EM) fields. In many cases, numerical methods are required to solve Maxwell’s equations, either in the frequency or time domain. For the time domain, a 