Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations

Y. Xu, Chi-Wang Shu

    Research output: Contribution to journal › Article › Academic › peer-review

    64 Citations (Scopus)

    Abstract

    In this paper we develop a local discontinuous Galerkin method to solve the Kuramoto–Sivashinsky equations and the Ito-type coupled KdV equations. The L2 stability of the schemes is obtained for both of these nonlinear equations. We use both the traditional nonlinearly stable explicit high order Runge–Kutta methods and the explicit exponential time differencing method for the time discretization; the latter can achieve high order accuracy and maintain good stability while avoiding the very restrictive explicit stability limit of the former when the PDE contains higher order spatial derivatives. Numerical examples are shown to demonstrate the accuracy and capability of these methods.
    Original languageUndefined
    Article number10.1016/j.cma.2005.06.021
    Pages (from-to)3430-3447
    Number of pages18
    JournalComputer methods in applied mechanics and engineering
    Volume195
    Issue number06EX1521/25-28
    DOIs
    Publication statusPublished - 1 May 2006

    Keywords

    • EWI-8093
    • MSC-65M60
    • IR-63667
    • METIS-237590
    • MSC-35Q53

    Cite this

    Xu, Y. ; Shu, Chi-Wang. / Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations. In: Computer methods in applied mechanics and engineering. 2006 ; Vol. 195, No. 06EX1521/25-28. pp. 3430-3447.
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    title = "Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations",
    abstract = "In this paper we develop a local discontinuous Galerkin method to solve the Kuramoto–Sivashinsky equations and the Ito-type coupled KdV equations. The L2 stability of the schemes is obtained for both of these nonlinear equations. We use both the traditional nonlinearly stable explicit high order Runge–Kutta methods and the explicit exponential time differencing method for the time discretization; the latter can achieve high order accuracy and maintain good stability while avoiding the very restrictive explicit stability limit of the former when the PDE contains higher order spatial derivatives. Numerical examples are shown to demonstrate the accuracy and capability of these methods.",
    keywords = "EWI-8093, MSC-65M60, IR-63667, METIS-237590, MSC-35Q53",
    author = "Y. Xu and Chi-Wang Shu",
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    year = "2006",
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    language = "Undefined",
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    pages = "3430--3447",
    journal = "Computer methods in applied mechanics and engineering",
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    Xu, Y & Shu, C-W 2006, 'Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations', Computer methods in applied mechanics and engineering, vol. 195, no. 06EX1521/25-28, 10.1016/j.cma.2005.06.021, pp. 3430-3447. https://doi.org/10.1016/j.cma.2005.06.021

    Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations. / Xu, Y.; Shu, Chi-Wang.

    In: Computer methods in applied mechanics and engineering, Vol. 195, No. 06EX1521/25-28, 10.1016/j.cma.2005.06.021, 01.05.2006, p. 3430-3447.

    Research output: Contribution to journal › Article › Academic › peer-review

    TY - JOUR

    T1 - Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations

    AU - Xu, Y.

    AU - Shu, Chi-Wang

    N1 - 10.1016/j.cma.2005.06.021

    PY - 2006/5/1

    Y1 - 2006/5/1

    N2 - In this paper we develop a local discontinuous Galerkin method to solve the Kuramoto–Sivashinsky equations and the Ito-type coupled KdV equations. The L2 stability of the schemes is obtained for both of these nonlinear equations. We use both the traditional nonlinearly stable explicit high order Runge–Kutta methods and the explicit exponential time differencing method for the time discretization; the latter can achieve high order accuracy and maintain good stability while avoiding the very restrictive explicit stability limit of the former when the PDE contains higher order spatial derivatives. Numerical examples are shown to demonstrate the accuracy and capability of these methods.

    AB - In this paper we develop a local discontinuous Galerkin method to solve the Kuramoto–Sivashinsky equations and the Ito-type coupled KdV equations. The L2 stability of the schemes is obtained for both of these nonlinear equations. We use both the traditional nonlinearly stable explicit high order Runge–Kutta methods and the explicit exponential time differencing method for the time discretization; the latter can achieve high order accuracy and maintain good stability while avoiding the very restrictive explicit stability limit of the former when the PDE contains higher order spatial derivatives. Numerical examples are shown to demonstrate the accuracy and capability of these methods.

    KW - EWI-8093

    KW - MSC-65M60

    KW - IR-63667

    KW - METIS-237590

    KW - MSC-35Q53

    U2 - 10.1016/j.cma.2005.06.021

    DO - 10.1016/j.cma.2005.06.021

    M3 - Article

    VL - 195

    SP - 3430

    EP - 3447

    JO - Computer methods in applied mechanics and engineering

    JF - Computer methods in applied mechanics and engineering

    SN - 0045-7825

    IS - 06EX1521/25-28

    M1 - 10.1016/j.cma.2005.06.021

    ER -

    Xu Y, Shu C-W. Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations. Computer methods in applied mechanics and engineering. 2006 May 1;195(06EX1521/25-28):3430-3447. 10.1016/j.cma.2005.06.021. https://doi.org/10.1016/j.cma.2005.06.021

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