Discontinuous Hamiltonian finite element method for linear hyperbolic systems

Y. Xu, Jacobus J.W. van der Vegt, Onno Bokhove

    Research output: Contribution to journalArticleAcademicpeer-review

    4 Citations (Scopus)
    46 Downloads (Pure)

    Abstract

    We develop a Hamiltonian discontinuous finite element discretization of a generalized Hamiltonian system for linear hyperbolic systems, which include the rotating shallow water equations, the acoustic and Maxwell equations. These equations have a Hamiltonian structure with a bilinear Poisson bracket, and as a consequence the phase-space structure, “mass��? and energy are preserved. We discretize the bilinear Poisson bracket in each element with discontinuous elements and introduce numerical fluxes via integration by parts while preserving the skew-symmetry of the bracket. This automatically results in a mass and energy conservative discretization. When combined with a symplectic time integration method, energy is approximately conserved and shows no drift. For comparison, the discontinuous Galerkin method for this problem is also used. A variety numerical examples is shown to illustrate the accuracy and capability of the new method.
    Original languageUndefined
    Article number10.1007/s10915-008-9191-y
    Pages (from-to)241-265
    Number of pages25
    JournalJournal of scientific computing
    Volume35
    Issue numberWP 08-02/2-3
    DOIs
    Publication statusPublished - 2008

    Keywords

    • Rotating shallow water equations · Acoustic equations · Maxwell equations ·Hamiltonian dynamics · Discontinuous Galerkin method · Numerical flux
    • IR-65304
    • METIS-255124
    • EWI-14888

    Cite this

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    title = "Discontinuous Hamiltonian finite element method for linear hyperbolic systems",
    abstract = "We develop a Hamiltonian discontinuous finite element discretization of a generalized Hamiltonian system for linear hyperbolic systems, which include the rotating shallow water equations, the acoustic and Maxwell equations. These equations have a Hamiltonian structure with a bilinear Poisson bracket, and as a consequence the phase-space structure, “mass��? and energy are preserved. We discretize the bilinear Poisson bracket in each element with discontinuous elements and introduce numerical fluxes via integration by parts while preserving the skew-symmetry of the bracket. This automatically results in a mass and energy conservative discretization. When combined with a symplectic time integration method, energy is approximately conserved and shows no drift. For comparison, the discontinuous Galerkin method for this problem is also used. A variety numerical examples is shown to illustrate the accuracy and capability of the new method.",
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    author = "Y. Xu and {van der Vegt}, {Jacobus J.W.} and Onno Bokhove",
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    year = "2008",
    doi = "10.1007/s10915-008-9191-y",
    language = "Undefined",
    volume = "35",
    pages = "241--265",
    journal = "Journal of scientific computing",
    issn = "0885-7474",
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    Discontinuous Hamiltonian finite element method for linear hyperbolic systems. / Xu, Y.; van der Vegt, Jacobus J.W.; Bokhove, Onno.

    In: Journal of scientific computing, Vol. 35, No. WP 08-02/2-3, 10.1007/s10915-008-9191-y, 2008, p. 241-265.

    Research output: Contribution to journalArticleAcademicpeer-review

    TY - JOUR

    T1 - Discontinuous Hamiltonian finite element method for linear hyperbolic systems

    AU - Xu, Y.

    AU - van der Vegt, Jacobus J.W.

    AU - Bokhove, Onno

    N1 - 10.1007/s10915-008-9191-y

    PY - 2008

    Y1 - 2008

    N2 - We develop a Hamiltonian discontinuous finite element discretization of a generalized Hamiltonian system for linear hyperbolic systems, which include the rotating shallow water equations, the acoustic and Maxwell equations. These equations have a Hamiltonian structure with a bilinear Poisson bracket, and as a consequence the phase-space structure, “mass��? and energy are preserved. We discretize the bilinear Poisson bracket in each element with discontinuous elements and introduce numerical fluxes via integration by parts while preserving the skew-symmetry of the bracket. This automatically results in a mass and energy conservative discretization. When combined with a symplectic time integration method, energy is approximately conserved and shows no drift. For comparison, the discontinuous Galerkin method for this problem is also used. A variety numerical examples is shown to illustrate the accuracy and capability of the new method.

    AB - We develop a Hamiltonian discontinuous finite element discretization of a generalized Hamiltonian system for linear hyperbolic systems, which include the rotating shallow water equations, the acoustic and Maxwell equations. These equations have a Hamiltonian structure with a bilinear Poisson bracket, and as a consequence the phase-space structure, “mass��? and energy are preserved. We discretize the bilinear Poisson bracket in each element with discontinuous elements and introduce numerical fluxes via integration by parts while preserving the skew-symmetry of the bracket. This automatically results in a mass and energy conservative discretization. When combined with a symplectic time integration method, energy is approximately conserved and shows no drift. For comparison, the discontinuous Galerkin method for this problem is also used. A variety numerical examples is shown to illustrate the accuracy and capability of the new method.

    KW - Rotating shallow water equations · Acoustic equations · Maxwell equations ·Hamiltonian dynamics · Discontinuous Galerkin method · Numerical flux

    KW - IR-65304

    KW - METIS-255124

    KW - EWI-14888

    U2 - 10.1007/s10915-008-9191-y

    DO - 10.1007/s10915-008-9191-y

    M3 - Article

    VL - 35

    SP - 241

    EP - 265

    JO - Journal of scientific computing

    JF - Journal of scientific computing

    SN - 0885-7474

    IS - WP 08-02/2-3

    M1 - 10.1007/s10915-008-9191-y

    ER -