HOS Simulations of Nonlinear Water Waves in Complex Media

  • Philippe GuyenneEmail author
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)


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.


Bathymetry Dirichlet–Neumann operator Sea ice Series expansion Spectral method Vorticity Water waves 

Mathematics Subject Classification (2000)

Primary 76B15; Secondary 65M70 



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. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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