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HOS Simulations of Nonlinear Water Waves in Complex Media

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

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.

Keywords

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

Mathematics Subject Classification (2000)

Primary 76B15; Secondary 65M70 

Notes

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”.

References

  1. 1.
    L. af Klinteberg, A.K. Tornberg, A fast integral equation method for solid particles in viscous flow using quadrature by expansion. J. Comput. Phys. 326, 420–445 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. Cathala, Asymptotic shallow water models with non smooth topographies. Monatsh. Math. 179, 325–353 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    R. Coifman, Y. Meyer, Nonlinear harmonic analysis and analytic dependence. Proc. Symp. Pure Math. 43, 71–78 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Compelli, Hamiltonian formulation of 2 bounded immiscible media with constant non-zero vorticities and a common interface. Wave Motion 54, 115–124 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    W. Craig, D.P. Nicholls, Traveling gravity water waves in two and three dimensions. Eur. J. Mech. B/Fluids 21, 615–641 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    W. Craig, C. Sulem, Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83 (1993)MathSciNetCrossRefGoogle Scholar
  7. 7.
    W. Craig, P. Guyenne, H. Kalisch, Hamiltonian long wave expansions for free surfaces and interfaces. Commun. Pure Appl. Math. 58, 1587–1641 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    W. Craig, P. Guyenne, D.P. Nicholls, C. Sulem, Hamiltonian long-wave expansions for water waves over a rough bottom. Proc. R. Soc. A 461, 839–873 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    W. Craig, P. Guyenne, J. Hammack, D. Henderson, C. Sulem, Solitary water wave interactions. Phys. Fluids 18, 057106 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    W. Craig, P. Guyenne, C. Sulem, Water waves over a random bottom. J. Fluid Mech. 640, 79–107 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    W. Craig, P. Guyenne, C. Sulem, Internal waves coupled to surface gravity waves in three dimensions. Commun. Math. Sci. 13, 893–910 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    M.W. Dingemans, Comparison of computations with Boussinesq-like models and laboratory measurements, in Technical Report H1684.12 (Delft Hydraulics, Delft, 1994)Google Scholar
  13. 13.
    D.G. Dommermuth, D.K.P. Yue, A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267–288 (1987)CrossRefGoogle Scholar
  14. 14.
    G. Ducrozet, F. Bonnefoy, D. Le Touzé, P. Ferrant, HOS-ocean: open-source solver for nonlinear waves in open ocean based on high-order spectral method. Comput. Phys. Commun. 203, 245–254 (2016)CrossRefGoogle Scholar
  15. 15.
    M. Francius, C. Kharif, S. Viroulet,Nonlinear simulations of surface waves in finite depth on a linear shear current, in Proceedings of the 7th International Conference on Coastal Dynamics (2013), pp. 649–660.Google Scholar
  16. 16.
    L. Greengard, V. Rokhlin, A fast algorithm for particle simulations. J. Comput. Phys. 73, 325–348 (1987)MathSciNetCrossRefGoogle Scholar
  17. 17.
    S.T. Grilli, P. Guyenne, F. Dias, A fully nonlinear model for three-dimensional overturning waves over an arbitrary bottom. Int. J. Numer. Meth. Fluids 35, 829–867 (2001)CrossRefGoogle Scholar
  18. 18.
    P. Guyenne, A high-order spectral method for nonlinear water waves in the presence of a linear shear current. Comput. Fluids 154, 224–235 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    P. Guyenne, S.T. Grilli, Numerical study of three-dimensional overturning waves in shallow water. J. Fluid Mech. 547, 361–388 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    P. Guyenne, D.P. Nicholls, Numerical simulation of solitary waves on plane slopes. Math. Comput. Simul. 69, 269–281 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    P. Guyenne, D.P. Nicholls, A high-order spectral method for nonlinear water waves over moving bottom topography. SIAM J. Sci. Comput. 30, 81–101 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    P. Guyenne, E.I. Părău, Computations of fully nonlinear hydroelastic solitary waves on deep water. J. Fluid Mech. 713, 307–329 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    P. Guyenne, E.I. Părău, Finite-depth effects on solitary waves in a floating ice sheet. J. Fluids Struct. 49, 242–262 (2014)CrossRefGoogle Scholar
  24. 24.
    P. Guyenne, E.I. Părău, Numerical study of solitary wave attenuation in a fragmented ice sheet. Phys. Rev. Fluids 2, 034002 (2017)CrossRefGoogle Scholar
  25. 25.
    P. Guyenne, D. Lannes, J.-C. Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves. Nonlinearity 23, 237–275 (2010)MathSciNetCrossRefGoogle Scholar
  26. 26.
    N. Hale, A. Townsend, A fast, simple, and stable Chebyshev–Legendre transform using an asymptotic formula. SIAM J. Sci. Comput. 36, A148–A167 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    T.Y. Hou, J.S. Lowengrub, M.J. Shelley, Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114, 312–338 (1994)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Y. Liu, D.K.P. Yue, On generalized Bragg scattering of surface waves by bottom ripples. J. Fluid Mech. 356, 297–326 (1998)MathSciNetCrossRefGoogle Scholar
  29. 29.
    P.A. Milewski, Z. Wang, Three dimensional flexural-gravity waves. Stud. Appl. Math. 131, 135–148 (2013)MathSciNetCrossRefGoogle Scholar
  30. 30.
    D.P. Nicholls, Traveling water waves: spectral continuation methods with parallel implementation. J. Comput. Phys. 143, 224–240 (1998)MathSciNetCrossRefGoogle Scholar
  31. 31.
    D.P. Nicholls, Boundary perturbation methods for water waves. GAMM-Mitt. 30, 44–74 (2007)MathSciNetCrossRefGoogle Scholar
  32. 32.
    D.P. Nicholls, F. Reitich, Stability of high-order perturbative methods for the computation of Dirichlet–Neumann operators. J. Comput. Phys. 170, 276–298 (2001)MathSciNetCrossRefGoogle Scholar
  33. 33.
    D.P. Nicholls, F. Reitich, A new approach to analyticity of Dirichlet–Neumann operators. Proc. Roy. Soc. Edinburgh Sect. A 131, 1411–1433 (2001)MathSciNetCrossRefGoogle Scholar
  34. 34.
    D.P. Nicholls, M. Taber, Joint analyticity and analytic continuation of Dirichlet–Neumann operators on doubly perturbed domains. J. Math. Fluid Mech. 10, 238–271 (2008)MathSciNetCrossRefGoogle Scholar
  35. 35.
    P.I. Plotnikov, J.F. Toland, Modelling nonlinear hydroelastic waves. Phil. Trans. R. Soc. Lond. A 369, 2942–2956 (2011)MathSciNetCrossRefGoogle Scholar
  36. 36.
    P. Wadhams, V.A. Squire, D.J. Goodman, A.M. Cowan, S.C. Moore, The attenuation rates of ocean waves in the marginal ice zone. J. Geophys. Res. 93, 6799–6818 (1988)CrossRefGoogle Scholar
  37. 37.
    E. Wahlén, A Hamiltonian formulation of water waves with constant vorticity. Lett. Math. Phys. 79, 303–315 (2007)MathSciNetCrossRefGoogle Scholar
  38. 38.
    B.J. West, K.A. Brueckner, R.S. Janda, D.M. Milder, R.L. Milton, A new numerical method for surface hydrodynamics. J. Geophys. Res. 92, 11803–11824 (1987)CrossRefGoogle Scholar
  39. 39.
    L. Xu, P. Guyenne, Numerical simulation of three-dimensional nonlinear water waves. J. Comput. Phys. 228, 8446–8466 (2009)MathSciNetCrossRefGoogle Scholar
  40. 40.
    V.E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194 (1968)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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