### 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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Place of Publication | Amsterdam |

Publisher | Centrum voor Wiskunde en Informatica |

Number of pages | 26 |

Publication status | Published - Sep 2008 |

### Publication series

Name | Modelling, Analysis and Simulation |
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No. | MAS-R0806 |

ISSN (Print) | 1386-3703 |

### Keywords

- IR-68838
- MSC-65L20
- EWI-16956
- MSC-65L05
- MSC-65M12
- METIS-264215
- MSC-65M20

## Cite this

Verwer, J. G., & Bochev, M. A. (2008).

*Unconditionally stable integration of Maxwell's equations*. (Modelling, Analysis and Simulation; No. MAS-R0806). Amsterdam: Centrum voor Wiskunde en Informatica.