Unconditionally stable integration of Maxwell's equations

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Abstract

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 semi-discrete 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. © 2009 Elsevier Inc. All rights reserved.

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Verwer, J. G., & Botchev, M. A. (2009). Unconditionally stable integration of Maxwell’s equations. Linear Algebra and Its Applications, 431(3–4), 300–317. https://doi.org/10.1016/j.laa.2008.12.036

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