A (Dis)continuous finite element model for generalized 2D vorticity dynamics

E. Bernsen, Onno Bokhove, Jacobus J.W. van der Vegt

    Research output: Book/ReportReportProfessional

    55 Downloads (Pure)

    Abstract

    A mixed continuous and discontinuous Galerkin finite element discretization is constructed for a generalized vorticity streamfunction formulation in two spatial dimensions. This formulation consists of a hyperbolic (potential) vorticity equation and a linear elliptic equation for a (transport) streamfunction. The generalized formulation includes three systems in geophysical fluid dynamics: the incompressible Euler equations, the barotropic quasi-geostrophic equations and the rigid-lid equations. Multiple connected domains are considered with impenetrable and curved boundaries such that the circulation at each connected piece of boundary must be introduced. The generalized system is shown to globally conserve energy and weighted smooth functions of the vorticity. In particular, the weighted square vorticity or enstrophy is conserved. By construction, the spatial finite-element discretization is shown to conserve energy and is $L^2$-stable in the enstrophy norm. The method is verified by numerical experiments which support our error estimates. Particular attention is paid to match the continuous and discontinuous discretization. Hence, the implementation with a third-order Runge-Kutta time discretization conserves energy and is $L^2$-stable in the enstrophy norm for increasing time resolution in multiple connected curved domains.
    Original languageUndefined
    Place of PublicationEnschede
    PublisherToegepaste Wiskunde
    ISBN (Print)169-2690
    Publication statusPublished - 2005

    Publication series

    NameMemoranda
    PublisherDepartment of Applied Mathematics, University of Twente
    No.1787
    ISSN (Print)0169-2690

    Keywords

    • EWI-3607
    • METIS-227276
    • IR-65971
    • MSC-35M10
    • MSC-65M15

    Cite this

    Bernsen, E., Bokhove, O., & van der Vegt, J. J. W. (2005). A (Dis)continuous finite element model for generalized 2D vorticity dynamics. (Memoranda; No. 1787). Enschede: Toegepaste Wiskunde.
    Bernsen, E. ; Bokhove, Onno ; van der Vegt, Jacobus J.W. / A (Dis)continuous finite element model for generalized 2D vorticity dynamics. Enschede : Toegepaste Wiskunde, 2005. (Memoranda; 1787).
    @book{40092e26f661414fac00857d0da7674e,
    title = "A (Dis)continuous finite element model for generalized 2D vorticity dynamics",
    abstract = "A mixed continuous and discontinuous Galerkin finite element discretization is constructed for a generalized vorticity streamfunction formulation in two spatial dimensions. This formulation consists of a hyperbolic (potential) vorticity equation and a linear elliptic equation for a (transport) streamfunction. The generalized formulation includes three systems in geophysical fluid dynamics: the incompressible Euler equations, the barotropic quasi-geostrophic equations and the rigid-lid equations. Multiple connected domains are considered with impenetrable and curved boundaries such that the circulation at each connected piece of boundary must be introduced. The generalized system is shown to globally conserve energy and weighted smooth functions of the vorticity. In particular, the weighted square vorticity or enstrophy is conserved. By construction, the spatial finite-element discretization is shown to conserve energy and is $L^2$-stable in the enstrophy norm. The method is verified by numerical experiments which support our error estimates. Particular attention is paid to match the continuous and discontinuous discretization. Hence, the implementation with a third-order Runge-Kutta time discretization conserves energy and is $L^2$-stable in the enstrophy norm for increasing time resolution in multiple connected curved domains.",
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    author = "E. Bernsen and Onno Bokhove and {van der Vegt}, {Jacobus J.W.}",
    note = "Imported from MEMORANDA",
    year = "2005",
    language = "Undefined",
    isbn = "169-2690",
    series = "Memoranda",
    publisher = "Toegepaste Wiskunde",
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    Bernsen, E, Bokhove, O & van der Vegt, JJW 2005, A (Dis)continuous finite element model for generalized 2D vorticity dynamics. Memoranda, no. 1787, Toegepaste Wiskunde, Enschede.

    A (Dis)continuous finite element model for generalized 2D vorticity dynamics. / Bernsen, E.; Bokhove, Onno; van der Vegt, Jacobus J.W.

    Enschede : Toegepaste Wiskunde, 2005. (Memoranda; No. 1787).

    Research output: Book/ReportReportProfessional

    TY - BOOK

    T1 - A (Dis)continuous finite element model for generalized 2D vorticity dynamics

    AU - Bernsen, E.

    AU - Bokhove, Onno

    AU - van der Vegt, Jacobus J.W.

    N1 - Imported from MEMORANDA

    PY - 2005

    Y1 - 2005

    N2 - A mixed continuous and discontinuous Galerkin finite element discretization is constructed for a generalized vorticity streamfunction formulation in two spatial dimensions. This formulation consists of a hyperbolic (potential) vorticity equation and a linear elliptic equation for a (transport) streamfunction. The generalized formulation includes three systems in geophysical fluid dynamics: the incompressible Euler equations, the barotropic quasi-geostrophic equations and the rigid-lid equations. Multiple connected domains are considered with impenetrable and curved boundaries such that the circulation at each connected piece of boundary must be introduced. The generalized system is shown to globally conserve energy and weighted smooth functions of the vorticity. In particular, the weighted square vorticity or enstrophy is conserved. By construction, the spatial finite-element discretization is shown to conserve energy and is $L^2$-stable in the enstrophy norm. The method is verified by numerical experiments which support our error estimates. Particular attention is paid to match the continuous and discontinuous discretization. Hence, the implementation with a third-order Runge-Kutta time discretization conserves energy and is $L^2$-stable in the enstrophy norm for increasing time resolution in multiple connected curved domains.

    AB - A mixed continuous and discontinuous Galerkin finite element discretization is constructed for a generalized vorticity streamfunction formulation in two spatial dimensions. This formulation consists of a hyperbolic (potential) vorticity equation and a linear elliptic equation for a (transport) streamfunction. The generalized formulation includes three systems in geophysical fluid dynamics: the incompressible Euler equations, the barotropic quasi-geostrophic equations and the rigid-lid equations. Multiple connected domains are considered with impenetrable and curved boundaries such that the circulation at each connected piece of boundary must be introduced. The generalized system is shown to globally conserve energy and weighted smooth functions of the vorticity. In particular, the weighted square vorticity or enstrophy is conserved. By construction, the spatial finite-element discretization is shown to conserve energy and is $L^2$-stable in the enstrophy norm. The method is verified by numerical experiments which support our error estimates. Particular attention is paid to match the continuous and discontinuous discretization. Hence, the implementation with a third-order Runge-Kutta time discretization conserves energy and is $L^2$-stable in the enstrophy norm for increasing time resolution in multiple connected curved domains.

    KW - EWI-3607

    KW - METIS-227276

    KW - IR-65971

    KW - MSC-35M10

    KW - MSC-65M15

    M3 - Report

    SN - 169-2690

    T3 - Memoranda

    BT - A (Dis)continuous finite element model for generalized 2D vorticity dynamics

    PB - Toegepaste Wiskunde

    CY - Enschede

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    Bernsen E, Bokhove O, van der Vegt JJW. A (Dis)continuous finite element model for generalized 2D vorticity dynamics. Enschede: Toegepaste Wiskunde, 2005. (Memoranda; 1787).