### Abstract

This article deals with time integration for the second-order Maxwell equations with possibly non-zero conductivity in the context of the discontinuous Galerkin finite element method DG-FEM) and the $H(\mathrm{curl})$-conforming FEM. For the spatial discretisation, hierarchic $H(\mathrm{curl})$-conforming basis functions are used up to polynomial order $p=3$ over tetrahedral meshes, meaning fourth-order convergence rate. A high-order polynomial basis often warrants the use of high-order time-integration schemes, but many well-known high-order schemes may suffer from a severe time-step stability restriction owing to the conductivity term. We investigate several possible time-integration methods from the point of view of accuracy, stability and computational work. We also carry out a numerical Fourier analysis to study the dispersion and dissipation properties of the semi-discrete DG-FEM scheme as well as the fully-discrete schemes with several of the time-integration methods. The dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps.

Original language | Undefined |
---|---|

Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 25 |

Publication status | Published - Feb 2012 |

### Publication series

Name | Memorandum / Department of Applied Mathematics |
---|---|

Publisher | University of Twente, Department of Applied Mathematics |

No. | 1975 |

ISSN (Print) | 1874-4850 |

ISSN (Electronic) | 1874-4850 |

### Keywords

- Second-order damped Maxwell wave equation
- High-order numerical time integration
- H(curl)-conforming finite element method
- IR-79682
- METIS-285242
- Discontinuous Galerkin finite element method
- MSC-65M60
- MSC-65M20
- MSC-65L20
- MSC-65L06
- EWI-21520

## Cite this

Sarmany, D., Bochev, M. A., & van der Vegt, J. J. W. (2012).

*Time-integration methods for finite element discretisations of the second-order Maxwell equation*. (Memorandum / Department of Applied Mathematics; No. 1975). Enschede: University of Twente, Department of Applied Mathematics.