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On Recent Numerical Methods for Steady Periodic Water Waves

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Nonlinear Water Waves

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Abstract

The study of steady periodic water waves, analytically as well as numerically, is a very active field of research. We describe some of the more recent numerical approaches to computing these waves numerically as well as the corresponding results. The focus of this work is on the different formulations as well as their limitations and similarities.

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Notes

  1. 1.

    A condition for such a choice which ensures existence of solution is given by (1.6) in Theorem 1.1 of [8].

References

  1. M. Ablowitz, A.S. Fokas, Z. Musslimani, On a new non-local formulation of water waves. J. Fluid Mech. 562, 313–343 (2006)

    Article  MathSciNet  Google Scholar 

  2. D. Amann, K. Kalimeris, Numerical approximation of water waves through a deterministic algorithm. J. Math. Fluid Mech. 20, 1815–1833 (2018)

    Article  MathSciNet  Google Scholar 

  3. D. Amann, K. Kalimeris, A numerical continuation approach for computing water waves of large wave height. Eur. J. Mech. B/Fluids 67, 314–328 (2018)

    Article  MathSciNet  Google Scholar 

  4. A.C. Ashton, A. Fokas, A non-local formulation of rotational water waves. J. Fluid Mech. 689, 129–148 (2011)

    Article  MathSciNet  Google Scholar 

  5. W. Choi, Nonlinear surface waves interacting with a linear shear current. Mat. Comput. Simul. 80(1), 29–36 (2009)

    Article  MathSciNet  Google Scholar 

  6. A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, vol. 81 (SIAM, Philadelphia, 2011)

    Book  Google Scholar 

  7. A. Constantin, On the modelling of equatorial waves. Geophys. Res. Lett. 39(5), L05602 (2012)

    Google Scholar 

  8. A. Constantin, W. Strauss, Exact steady periodic water waves with vorticity. Commun. Pure Appl. Math. 57(4), 481–527 (2004)

    Article  MathSciNet  Google Scholar 

  9. A. Constantin, K. Kalimeris, O. Scherzer, Approximations of steady periodic water waves in flows with constant vorticity. Nonlinear Anal. Real World Appl. 25, 276–306 (2015)

    Article  MathSciNet  Google Scholar 

  10. A.T. Da Silva, D. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281–302 (1988)

    Article  MathSciNet  Google Scholar 

  11. B. Deconinick, K. Oliveras, The instability of periodic surface gravity waves. J. Fluid Mech. 675, 141–167 (2011)

    Article  MathSciNet  Google Scholar 

  12. M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie. J. Math. Pures Appl. 13, 217-291 (1934)

    MATH  Google Scholar 

  13. M. Ehrnström, J. Escher, E. Wahlén, Steady water waves with multiple critical layers. SIAM J. Math. Anal. 43(3), 1436–1456 (2011)

    Article  MathSciNet  Google Scholar 

  14. M. Ehrnström, J. Escher, G. Villari, Steady water waves with multiple critical layers: interior dynamics. J. Math. Fluid Mech. 14(3), 407–419 (2012)

    Article  MathSciNet  Google Scholar 

  15. A.S. Fokas, A. Nachbin, Water waves over a variable bottom: a non-local formulation and conformal mappings. J. Fluid Mech. 695, 288–309 (2012)

    Article  MathSciNet  Google Scholar 

  16. D. Henry, Large amplitude steady periodic waves for fixed-depth rotational flows. Commun. Partial Differ. Equ. 38(6), 1015–1037 (2013)

    Article  MathSciNet  Google Scholar 

  17. V.M. Hur, Shallow water models with constant vorticity. Eur. J. Mech. B/Fluids 73, 170–179 (2019)

    Article  MathSciNet  Google Scholar 

  18. K. Kalimeris, Asymptotic expansions for steady periodic water waves in flows with constant vorticity. Nonlinear Anal. Real World Appl. 37, 182–212 (2017)

    Article  MathSciNet  Google Scholar 

  19. P. Karageorgis, Dispersion relation for water waves with non-constant vorticity. Eur. J. Mech. B/Fluids 34, 7–12 (2012)

    Article  MathSciNet  Google Scholar 

  20. J. Ko, W. Strauss, Effect of vorticity on steady water waves. J. Fluid Mech. 608, 197–215 (2008)

    Article  MathSciNet  Google Scholar 

  21. J. Ko, W. Strauss, Large-amplitude steady rotational water waves. Eur. J. Mech. B/Fluids 27(2), 96 – 109 (2008)

    Article  MathSciNet  Google Scholar 

  22. M. Longuet-Higgins, M. Tanaka, On the crest instabilities of steep surface waves. J. Fluid Mech. 336, 51–68 (1997)

    Article  MathSciNet  Google Scholar 

  23. D.V. Maklakov, Almost-highest gravity waves on water of finite depth. Eur. J. Appl. Math. 13(1), 67–93 (2002)

    Article  MathSciNet  Google Scholar 

  24. P.A. Milewski, J.-M. Vanden-Broeck, Z. Wang, Dynamics of steep two-dimensional gravity–capillary solitary waves. J. Fluid Mech. 664, 466–477 (2010)

    Article  MathSciNet  Google Scholar 

  25. K. Oliveras, V. Vasan, A new equation describing travelling water waves. J. Fluid Mech. 717, 514–522 (2013)

    Article  MathSciNet  Google Scholar 

  26. K.L. Oliveras, V. Vasan, B. Deconinck, D. Henderson, Recovering the water-wave profile from pressure measurements. SIAM J. Appl. Math. 72(3), 897–918 (2012)

    Article  MathSciNet  Google Scholar 

  27. R. Quirchmayr, On irrotational flows beneath periodic traveling equatorial waves. J. Math. Fluid Mech. 19(2), 283–304 (2017)

    Article  MathSciNet  Google Scholar 

  28. R. Ribeiro, P.A. Milewski, A. Nachbin, Flow structure beneath rotational water waves with stagnation points. J. Fluid Mech. 812, 792–814 (2017)

    Article  MathSciNet  Google Scholar 

  29. L. Schwartz, J. Fenton, Strongly nonlinear waves. Annu. Rev. Fluid Mech. 14(1), 39–60 (1982)

    Article  MathSciNet  Google Scholar 

  30. J.A. Simmen, P. Saffman, Steady deep-water waves on a linear shear current. Stud. Appl. Math. 73(1), 35–57 (1985)

    Article  MathSciNet  Google Scholar 

  31. M. Souli, J. Zolesio, A. Ouahsine, Shape optimization for non-smooth geometry in two dimensions. Comput. Methods Appl. Mech. Eng. 188(1), 109–119 (2000)

    Article  MathSciNet  Google Scholar 

  32. W. Strauss, Steady water waves. Bull. Am. Math. Soc. 47(4), 671–694 (2010)

    Article  MathSciNet  Google Scholar 

  33. J.F. Toland, On the existence of a wave of greatest height and stokes’s conjecture. Proc. R. Soc. Lond. A Math. Phys. Sci. 363(1715), 469–485 (1978)

    Article  MathSciNet  Google Scholar 

  34. V. Vasan, K. Oliveras, Pressure beneath a traveling wave with constant vorticity. Discrete Contin. Dynam. Syst. 34, 3219–3239 (2014)

    Article  MathSciNet  Google Scholar 

  35. E. Wahlén, Steady water waves with a critical layer. J. Differ. Equ. 246(6), 2468–2483 (2009)

    Article  MathSciNet  Google Scholar 

  36. V.E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9(2), 190–194 (1968)

    Article  Google Scholar 

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Acknowledgements

The author was supported by the project Computation of large amplitude water waves (P 27755-N25), funded by the Austrian Science Fund (FWF). The author would like to thank the reviewers for their suggestions and comments, as those led to a more coherent and precise manuscript.

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria

    Dominic Amann

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  1. Dominic Amann
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Correspondence to Dominic Amann .

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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Amann, D. (2019). On Recent Numerical Methods for Steady Periodic Water Waves. 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_9

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