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Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth

Published online by Cambridge University Press:  06 April 2018

Didier Clamond
Affiliation:
Université Côte d’Azur, CNRS-LJAD UMR 7351, Parc Valrose, F-06108 Nice, France
Denys Dutykh
Affiliation:
Université Savoie Mont Blanc, CNRS-LAMA UMR 5127, Campus Scientifique, F-73376 Le Bourget-du-Lac, France
Corresponding
E-mail address:

Abstract

This paper describes an efficient algorithm for computing steady two-dimensional surface gravity waves in irrotational motion. The algorithm complexity is $O(N\log N)$ , $N$ being the number of Fourier modes. This feature allows the arbitrary precision computation of waves in arbitrary depth, i.e. it works efficiently for Stokes, cnoidal and solitary waves, even for quite large steepnesses, up to approximately 99 % of the maximum steepness for all wavelengths. In particular, the possibility to compute very long (cnoidal) waves accurately is a feature not shared by other algorithms and asymptotic expansions. The method is based on conformal mapping, the Babenko equation rewritten in a suitable way, the pseudo-spectral method and Petviashvili iterations. The efficiency of the algorithm is illustrated via some relevant numerical examples. The code is open source, so interested readers can easily check the claims, use and modify the algorithm.

Type
JFM Papers
Copyright
© 2018 Cambridge University Press 

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References

Ablowitz, M. J. & Musslimani, Z. H. 2005 Spectral renormalization method for computing self-localized solutions to nonlinear systems. Opt. Lett. 30, 21402142.10.1364/OL.30.002140CrossRefGoogle ScholarPubMed
Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Advanpix2017 Multiprecision Computing Toolbox for MATLAB. Advanpix LLC.Google Scholar
Álvarez, J. & Duràn, A. 2014a An extended Petviashvilli method for the numerical generation of traveling and localized waves. Commun. Nonlinear Sci. Numer. Simul. 19, 22722283.10.1016/j.cnsns.2013.12.004CrossRefGoogle Scholar
Álvarez, J. & Duràn, A. 2014b Petviashvilli type methods for traveling wave computations: I. Analysis of convergence. J. Comput. Appl. Maths 266, 3951.10.1016/j.cam.2014.01.015CrossRefGoogle Scholar
Babenko, K. I. 1987 Some remarks on the theory of surface waves of finite amplitude. Sov. Math. Dokl. 35, 599603.Google Scholar
Boyd, J. P. 2000 Chebyshev and Fourier Spectral Methods. Dover.Google Scholar
Byatt-Smith, J. G. B. 2001 Numerical solution of Nekrasov’s equation in the boundary layer near the crest for waves near the maximum height. Stud. Appl. Maths 106, 393405.10.1111/1467-9590.00171CrossRefGoogle Scholar
Clamond, D. 1999 Steady finite-amplitude waves on a horizontal seabed of arbitrary depth. J. Fluid Mech. 398, 4560.10.1017/S0022112099006151CrossRefGoogle Scholar
Clamond, D. 2003 Cnoidal-type surface waves in deep water. J. Fluid Mech. 489, 101120.10.1017/S0022112003005111CrossRefGoogle Scholar
Clamond, D. 2012 Note on the velocity and related fields of steady irrotational two-dimensional surface gravity waves. Phil. Trans. R. Soc. Lond. A 370 (1964), 15721586.10.1098/rsta.2011.0470CrossRefGoogle ScholarPubMed
Clamond, D. 2017a Remarks on Bernoulli constants, gauge conditions and phase velocities in the context of water waves. Appl. Maths Lett. 74, 114120.10.1016/j.aml.2017.05.018CrossRefGoogle Scholar
Clamond, D. 2018 New exact relations for steady irrotational two-dimensional gravity and capillary surface waves. Phil. Trans. R. Soc. Lond. A 376 (2111), 20170220.10.1098/rsta.2017.0220CrossRefGoogle ScholarPubMed
Clamond, D. & Constantin, A. 2013 Recovery of steady periodic wave profiles from pressure measurements at the bed. J. Fluid Mech. 714, 463475.10.1017/jfm.2012.490CrossRefGoogle Scholar
Clamond, D. & Dutykh, D. 2013 Fast accurate computation of the fully nonlinear solitary surface gravity waves. Comput. Fluids 84, 3538.10.1016/j.compfluid.2013.05.010CrossRefGoogle Scholar
Clamond, D., Dutykh, D. & Durán, A. 2015 A plethora of generalised solitary gravity–capillary water waves. J. Fluid Mech. 784, 664680.10.1017/jfm.2015.616CrossRefGoogle Scholar
Constantin, A. 2011 Nonlinear water waves with applications to wave–current interactions and tsunamis. In CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81. SIAM-Society for Industrial and Applied Mathematics.Google Scholar
Constantin, A. 2012 Nonlinear water waves. Phil. Trans. R. Soc. Lond. A 370 (1964), 15011504.10.1098/rsta.2011.0594CrossRefGoogle ScholarPubMed
Constantin, A. 2013 Mean velocities in a Stokes wave. Arch. Rat. Mech. Anal. 207 (3), 907917.10.1007/s00205-012-0584-6CrossRefGoogle Scholar
Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary–gravity waves. Annu. Rev. Fluid Mech. 31, 301346.10.1146/annurev.fluid.31.1.301CrossRefGoogle Scholar
Domokos, G. & Holmes, P. 2003 On nonlinear boundary-value problems: ghosts, parasites and discretizations. Proc. R. Soc. Lond. A 459, 15351561.10.1098/rspa.2002.1091CrossRefGoogle Scholar
Dutykh, D. & Clamond, D. 2014 Efficient computation of steady solitary gravity waves. Wave Motion 51, 8699.10.1016/j.wavemoti.2013.06.007CrossRefGoogle Scholar
Dyachenko, S. A., Lushnikov, P. M. & Korotkevich, A. O. 2013 The complex singularity of a Stokes wave. JETP Lett. 98 (11), 675679.10.1134/S0021364013240077CrossRefGoogle Scholar
Fenton, J. D. 1988 The numerical solution of steady water wave problems. Comput. Geosci. 14, 357368.10.1016/0098-3004(88)90066-0CrossRefGoogle Scholar
Fenton, J. D. 1999 Numerical methods for nonlinear waves. Adv. Coast. Ocean Engng 5, 241324.10.1142/9789812797544_0005CrossRefGoogle Scholar
Fenton, J. D.2015 Use of the programs Fourier, Cnoidal and Stokes for steady waves. Tech. Rep., http://johndfenton.com/Steady-waves/Instructions.pdf.Google Scholar
Germain, J.-P.1967 Contribution à l’étude de la houle en eau peu profonde. PhD thesis, Université de Grenoble.Google Scholar
Groves, M. D. 2004 Steady water waves. J. Nonlinear Math. Phys. 11 (4), 435460.10.2991/jnmp.2004.11.4.2CrossRefGoogle Scholar
Korteweg, D. J. & de Vries, G. 1895 On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Phil. Mag. 39 (240), 422443.10.1080/14786449508620739CrossRefGoogle Scholar
Lakoba, T. I. & Yang, J. 2007 A generalized Petviashvilli iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity. J. Comput. Phys. 226, 16681692.10.1016/j.jcp.2007.06.009CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. Lond. A 342, 157174.Google Scholar
Longuet-Higgins, M. S. 1984 New integral relations for gravity waves of finite amplitude. J. Fluid Mech. 149, 205215.10.1017/S0022112084002615CrossRefGoogle Scholar
Lourakis, M. L. A. & Argyros, A. A. 2005 Is Levenberg–Marquardt the most efficient optimization algorithm for implementing bundle adjustment? In Proc. 10th IEEE Int. Conf. Comp. Vision (ICCV’05), vol. 2, pp. 15261531. IEEE.Google Scholar
Maklakov, D. V. 2002 Almost highest gravity waves on water of finite depth. Eur. J. Appl. Maths 13, 6793.10.1017/S0956792501004739CrossRefGoogle Scholar
Maklakov, D. V. & Petrov, A. G. 2015 On steady non-breaking downstream waves and the wave resistance. J. Fluid Mech. 776, 290315.10.1017/jfm.2015.331CrossRefGoogle Scholar
McCowan, J. 1891 On the solitary wave. Phil. Mag. 32 (194), 4558.10.1080/14786449108621390CrossRefGoogle Scholar
Okamoto, I. & Shōji, M. 2001 The Mathematical Theory of Permanent Progressive Water-waves, Adv. Ser. Nonlin. Dyn., vol. 20. World Scientific.10.1142/4547CrossRefGoogle Scholar
Pelinovsky, D. E. & Stepanyants, Y. A. 2004 Convergence of Petviashvili’s iteration method for numerical approximation of stationary solutions of nonlinear wave equations. SIAM J. Numer. Anal. 42 (3), 11101127.10.1137/S0036142902414232CrossRefGoogle Scholar
Petviashvili, V. I. 1976 Equation of an extraordinary soliton. Sov. J. Plasma Phys. 2 (3), 469472.Google Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 62 (3), 553578.10.1017/S0022112074000802CrossRefGoogle Scholar
Starr, V. T. 1947 Momentum and energy integrals for gravity waves of finite height. J. Mar. Res. 6, 175193.Google Scholar
Stokes, G. G. 1880 Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form. In G.G. Stokes Math. and Phys. Papers, vol. 1, pp. 225228. Cambridge University Press.Google Scholar
Strauss, W. A. 2010 Steady water waves. Bull. Am. Math. Soc. 47, 671694.10.1090/S0273-0979-2010-01302-1CrossRefGoogle Scholar
Tanaka, M. 1986 The stability of solitary waves. Phys. Fluids 29 (3), 650655.10.1063/1.865459CrossRefGoogle Scholar
Vanden-Broeck, J.-M. 2010 Gravity–Capillary Free-Surface Flows, Cambridge Monographs on Mechanics. Cambridge University Press.10.1017/CBO9780511730276CrossRefGoogle Scholar
Williams, J. M. 1985 Near-limiting gravity waves in water of finite depth. Phil. Trans. R. Soc. Lond. A 314 (1530), 139188.Google Scholar
Yang, J. 2010 Nonlinear Waves in Integrable and Nonintegrable Systems, SIAM Monographs on Mathematical Modeling and Computation, vol. 16. SIAM.10.1137/1.9780898719680CrossRefGoogle Scholar

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Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth
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