Time-integration methods for finite element discretisations of the second-order Maxwell equation

D. Sarmany, Mikhail A. Bochev, Jacobus J.W. van der Vegt

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    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 languageUndefined
    Place of PublicationEnschede
    PublisherUniversity of Twente, Department of Applied Mathematics
    Number of pages25
    Publication statusPublished - Feb 2012

    Publication series

    NameMemorandum / Department of Applied Mathematics
    PublisherUniversity 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.