Unconditionally stable integration of Maxwell's equations

J.G. Verwer, Mikhail A. Bochev

Research output: Contribution to journalArticleAcademicpeer-review

17 Citations (Scopus)

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 phi-functions. A three-space dimensional test problem is used for numerical assessment and comparison with an economical second order implicit-explicit integrator. We further pay attention to the Chebyshev series expansion for computing the exponential operator for skew-symmetric matrices.
Original languageUndefined
Article number10.1016/j.laa.2008.12.036
Pages (from-to)300-317
Number of pages26
JournalLinear algebra and its applications
Volume431
Issue number3-4
DOIs
Publication statusPublished - Jul 2009

Keywords

  • EWI-17315
  • MSC-65L05
  • MSC-65L20
  • MSC-65M12
  • MSC-65M20
  • IR-69760
  • Krylov subspace iteration
  • exponential integration
  • implicit integration
  • conjugate gradient iteration
  • METIS-264501
  • Maxwell's equations

Cite this

Verwer, J. G., & Bochev, M. A. (2009). Unconditionally stable integration of Maxwell's equations. Linear algebra and its applications, 431(3-4), 300-317. [10.1016/j.laa.2008.12.036]. https://doi.org/10.1016/j.laa.2008.12.036
Verwer, J.G. ; Bochev, Mikhail A. / Unconditionally stable integration of Maxwell's equations. In: Linear algebra and its applications. 2009 ; Vol. 431, No. 3-4. pp. 300-317.
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Verwer, JG & Bochev, MA 2009, 'Unconditionally stable integration of Maxwell's equations' Linear algebra and its applications, vol. 431, no. 3-4, 10.1016/j.laa.2008.12.036, pp. 300-317. https://doi.org/10.1016/j.laa.2008.12.036

Unconditionally stable integration of Maxwell's equations. / Verwer, J.G.; Bochev, Mikhail A.

In: Linear algebra and its applications, Vol. 431, No. 3-4, 10.1016/j.laa.2008.12.036, 07.2009, p. 300-317.

Research output: Contribution to journalArticleAcademicpeer-review

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T1 - Unconditionally stable integration of Maxwell's equations

AU - Verwer, J.G.

AU - Bochev, Mikhail A.

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N2 - 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 phi-functions. A three-space dimensional test problem is used for numerical assessment and comparison with an economical second order implicit-explicit integrator. We further pay attention to the Chebyshev series expansion for computing the exponential operator for skew-symmetric matrices.

AB - 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 phi-functions. A three-space dimensional test problem is used for numerical assessment and comparison with an economical second order implicit-explicit integrator. We further pay attention to the Chebyshev series expansion for computing the exponential operator for skew-symmetric matrices.

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KW - MSC-65M12

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KW - IR-69760

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KW - exponential integration

KW - implicit integration

KW - conjugate gradient iteration

KW - METIS-264501

KW - Maxwell's equations

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