Numerical integration of Maxwell’s equations is often based on explicit methods accepting a stability step size restriction. In literature evidence is given that there is also a need for unconditionally stable methods, as exemplified by the successful alternating direction implicit – finite difference time domain scheme. In this paper, we discuss unconditionally stable integration for a general semidiscrete Maxwell system allowing non-Cartesian space grids as encountered in finite-element discretizations. Such grids exclude the alternating direction implicit approach. Particular attention is given to the second-order trapezoidal rule implemented with preconditioned conjugate gradient iteration and to second-order exponential integration using Krylov subspace iteration for evaluating the arising ϕ-functions. A three-space dimensional test problem is used for numerical assessment and comparison with an economical second-order implicit–explicit integrator.

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Linear Algebra and its Applications
Computational Dynamics

Verwer, J.G, & Botchev, M.A. (2009). Unconditionally stable integration of Maxwell's equations. Linear Algebra and its Applications, 431, 300–317.