Unconditionally stable integration of Maxwell’s equations

• Published in: Linear algebra and its applications 2009-07, Vol.431 (3), p.300-317
• Main Authors: Verwer, J.G., Botchev, M.A.
• Format: Article
• Language: English
• Subjects: Algebra; Applied classical electromagnetism; Conjugate gradient iteration; Electromagnetic wave propagation, radiowave propagation; Electromagnetism; electron and ion optics; Exact sciences and technology; Exponential integration; Fundamental areas of phenomenology (including applications); Implicit integration; Krylov subspace iteration; Linear and multilinear algebra, matrix theory; Mathematics; Maxwell’s equations; Physics; Sciences and techniques of general use
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2008.12.036

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.