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HOS Simulations of Nonlinear Water Waves in Complex Media

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Nonlinear Water Waves

Abstract

We present an overview of recent extensions of the high-order spectral method of Craig and Sulem (J Comput Phys 108:73–83, 1993) to simulating nonlinear water waves in a complex environment. Under consideration are cases of wave propagation in the presence of fragmented sea ice, variable bathymetry and a vertically sheared current. Key components of this method, which apply to all three cases, include reduction of the full problem to a lower-dimensional system involving boundary variables alone, and a Taylor series representation of the Dirichlet–Neumann operator. This results in a very efficient and accurate numerical solver by using the fast Fourier transform. Two-dimensional simulations of unsteady wave phenomena are shown to illustrate the performance and versatility of this approach.

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Acknowledgements

The author acknowledges support by the NSF through grant number DMS-1615480. He is grateful to the Erwin Schrödinger International Institute for Mathematics and Physics for its hospitality during a visit in the fall 2017, and to the organizers of the workshop “Nonlinear Water Waves—an Interdisciplinary Interface”.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

    Philippe Guyenne

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  1. Philippe Guyenne
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Correspondence to Philippe Guyenne .

Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, University College Cork, Cork, Ireland

    David Henry

  2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK

    Konstantinos Kalimeris

  3. School of Mathematics, University of East Anglia, Norwich, UK

    Emilian I. Părău

  4. Department of Mathematics, University College London, London, UK

    Jean-Marc Vanden-Broeck

  5. Centre for Mathematical Sciences, Lund University, Lund, Sweden

    Erik Wahlén

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Guyenne, P. (2019). HOS Simulations of Nonlinear Water Waves in Complex Media. In: Henry, D., Kalimeris, K., Părău, E., Vanden-Broeck, JM., Wahlén, E. (eds) Nonlinear Water Waves . Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-33536-6_4

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