Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations.

*(English)*Zbl 1153.65097The authors investigate systems of hyperbolic partial differential equations including nonconservative products, i.e., the systems are not equivalent to conservation laws in general. To consider discontinuous solutions, a definition of the nonconservative products is applied according to G. Dal Maso, P. G. LeFloch and F. Murat [J. Math. Pures Appl., IX. Sér. 74, No. 6, 483–548 (1995; Zbl 0853.35068)]. The authors construct a Galerkin finite element method using space-time elements with arbitrary space dimension, where the space variables and the time variable are not distinguished. It follows that the technique reduces to conservative space-time methods if the underlying system is equivalent to a conservation law.

Necessary stabilising terms are included in the scheme. Nonlinear systems, which result from advancing in physical time, are solved by the introduction of a pseudo-time derivative and numerical integration to steady state. Finite element methods in space dimensions only represent a special case of this construction. The authors arrange a formula for the numerical flux corresponding to the method. Thereby, different paths can be chosen out of some class to connect left and right states at a discontinuity in phase space.

About two-thirds of the article consists in the discussion of test examples. The authors present figures of numerical solutions and tables demonstrating numerical errors with respect to the number of used elements. Firstly, several simulations of the shallow water equations are given for one space dimension, where the topography is included as an unknown in the system. Secondly, a test case of the shallow water equations in two space dimensions is involved.

Thirdly, some simulations using a depth-averaged two-phase flow model are presented in one space dimension. In the latter case, different paths are selected to connect the left and right states at discontinuities. The corresponding numerical solutions do not differ significantly in case of shocks, i.e., the solutions are not sensitive with respect to the choice of the path.

However, the numerical solutions are more sensitive according to the path in case of contact discontinuities.

Necessary stabilising terms are included in the scheme. Nonlinear systems, which result from advancing in physical time, are solved by the introduction of a pseudo-time derivative and numerical integration to steady state. Finite element methods in space dimensions only represent a special case of this construction. The authors arrange a formula for the numerical flux corresponding to the method. Thereby, different paths can be chosen out of some class to connect left and right states at a discontinuity in phase space.

About two-thirds of the article consists in the discussion of test examples. The authors present figures of numerical solutions and tables demonstrating numerical errors with respect to the number of used elements. Firstly, several simulations of the shallow water equations are given for one space dimension, where the topography is included as an unknown in the system. Secondly, a test case of the shallow water equations in two space dimensions is involved.

Thirdly, some simulations using a depth-averaged two-phase flow model are presented in one space dimension. In the latter case, different paths are selected to connect the left and right states at discontinuities. The corresponding numerical solutions do not differ significantly in case of shocks, i.e., the solutions are not sensitive with respect to the choice of the path.

However, the numerical solutions are more sensitive according to the path in case of contact discontinuities.

Reviewer: Roland Pulch (Wuppertal)

##### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

35L60 | First-order nonlinear hyperbolic equations |

35L65 | Hyperbolic conservation laws |

35L67 | Shocks and singularities for hyperbolic equations |

76M10 | Finite element methods applied to problems in fluid mechanics |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

76T10 | Liquid-gas two-phase flows, bubbly flows |

##### Keywords:

hyperbolic equations; nonconservative products; discontinuous Galerkin finite element methods; numerical fluxes; two-phase flows; shallow water equations; arbitrary Lagrangian Eulerian formulation; numerical examples##### Software:

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\textit{S. Rhebergen} et al., J. Comput. Phys. 227, No. 3, 1887--1922 (2008; Zbl 1153.65097)

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