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On the Galerkin/Finite-Element Method for the Serre Equations

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Abstract

A highly accurate numerical scheme is presented for the Serre system of partial differential equations, which models the propagation of dispersive shallow water waves in the fully-nonlinear regime. The fully-discrete scheme utilizes the Galerkin / finite-element method based on smooth periodic splines in space, and an explicit fourth-order Runge–Kutta method in time. Computations compared with exact solitary and cnoidal wave solutions show that the scheme achieves the optimal orders of accuracy in space and time. These computations also show that the stability of this scheme does not impose very restrictive conditions on the temporal stepsize. In addition, solitary, cnoidal, and dispersive shock waves are studied in detail using this numerical scheme for the Serre system and compared with the ‘classical’ Boussinesq system for small-amplitude shallow water waves. The results show that the interaction of solitary waves in the Serre system is more inelastic. The efficacy of the numerical scheme for modeling dispersive shocks is shown by comparison with asymptotic results. These results have application to the modeling of shallow water waves of intermediate or large amplitude.

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Acknowledgments

D. Dutykh would like to acknowledge the hospitality of UC Merced during his visit in April 2013 and the support from ERC under the research project ERC-2011-AdG 290562-MULTIWAVE. D. Mitsotakis would like to thank Prof. Mark Hoefer for his suggestions and his comments and for the fruitful discussions on dispersive waves.

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Authors and Affiliations

  1. School of Natural Sciences, University of California, Merced, 5200 North Lake Road, Merced, CA , 95343, USA

    Dimitrios Mitsotakis & Boaz Ilan

  2. School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

    Denys Dutykh

  3. LAMA, UMR 5127 CNRS, Université de Savoie, Campus Scientifique, 73376 , Le Bourget-du-Lac Cedex, France

    Denys Dutykh

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  1. Dimitrios Mitsotakis
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Correspondence to Dimitrios Mitsotakis.

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Mitsotakis, D., Ilan, B. & Dutykh, D. On the Galerkin/Finite-Element Method for the Serre Equations. J Sci Comput 61, 166–195 (2014). https://doi.org/10.1007/s10915-014-9823-3

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  • DOI: https://doi.org/10.1007/s10915-014-9823-3

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