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Lagrange–Galerkin method for unsteady free surface water waves

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Computing and Visualization in Science

Abstract

We present a new numerical technique to approximate solutions to unsteady free surface flows modelled by the two-dimensional shallow water equations. The method we propose in this paper consists of an Eulerian–Lagrangian splitting of the equations along the characteristic curves. The Lagrangian stage of the splitting is treated by a non-oscillatory modified method of characteristics, while the Eulerian stage is approximated by an implicit time integration scheme using finite element method for spatial discretization. The combined two stages lead to a Lagrange–Galerkin method which is robust, second order accurate, and simple to implement for problems on complex geometry. Numerical results are shown for several test problems with different ranges of difficulty.

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Authors and Affiliations

  1. FB Mathematik, Universität Kaiserslautern, 67663, Kaiserslautern, Germany

    Mohammed Seaïd

  2. Dpto. de Matemáticas y Física Aplicada y Ciencias de la Naturaleza, ESCET, Univ. Rey Juan Carlos, C/ Tulipán, s/n, 28933, Móstoles-Madrid, Spain

    Mofdi El-Amrani

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  1. Mohammed Seaïd
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  2. Mofdi El-Amrani
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Correspondence to Mohammed Seaïd.

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Communicated by G.Wittum

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Seaïd, M., El-Amrani, M. Lagrange–Galerkin method for unsteady free surface water waves. Comput. Visual Sci. 9, 209–228 (2006). https://doi.org/10.1007/s00791-006-0027-8

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  • DOI: https://doi.org/10.1007/s00791-006-0027-8

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