### 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 language | Undefined |
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Article number | 10.1016/j.laa.2008.12.036 |

Pages (from-to) | 300-317 |

Number of pages | 26 |

Journal | Linear algebra and its applications |

Volume | 431 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 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