Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations

S. Rhebergen, O. Bokhove, J.J.W. van der Vegt

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    We present space- and space-time discontinuous Galerkin finite element (DGFEM) formulations for systems containing nonconservative products, such as occur in dispersed multiphase flow equations. The main criterium we pose on the weak formulation is that if the system of nonconservative partial differential equations can be transformed into conservative form, then the formulation must reduce to that for conservative systems. Standard DGFEM formulations cannot be applied to nonconservative systems of partial differential equations. We therefore introduce the theory of weak solutions for nonconservative products into the DGFEM formulation leading to the new question how to define the path connecting left and right states across a discontinuity. The effect of different paths on the numerical solution is investigated and found to be small. We also introduce a new numerical flux that is able to deal with nonconservative products. Our scheme is applied to two different systems of partial differential equations. First, we consider the shallow water equations, where topography leads to nonconservative products, in which the known, possibly discontinuous, topography is formally taken as an unknown in the system. Second, we consider a simplification of a depth-averaged two-phase flow model which contains more intrinsic nonconservative products.
    Original languageEnglish
    Pages (from-to)1887-1922
    Number of pages36
    JournalJournal of computational physics
    Issue number3
    Publication statusPublished - 10 Jan 2008


    • PACS-02.60.Cb
    • PACS-02.70.Dh
    • MSC-35L60
    • MSC-35L65
    • MSC-35L67
    • Nonconservative products
    • Two-phase flows
    • Numerical fluxes
    • MSC-65M60
    • MSC-76M10
    • PACS-47.55.-t
    • PACS-47.85.Dh
    • Arbitrary Lagrangian Eulerian (ALE) formulation
    • Discontinuous Galerkin finite element methods


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