Numerical comparisons of two long-wave limit models

Stéphane Labbé; Lionel Paumond

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2004)

  • Volume: 38, Issue: 3, page 419-436
  • ISSN: 0764-583X

Abstract

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The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations 16 (2003) 1039–1064; Pego and Quintero, Physica D 132 (1999) 476–496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically the link between (KP) and (BL) and we point out the coupling effects emerging by considering two solitary waves propagating in two opposite directions.

How to cite

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Labbé, Stéphane, and Paumond, Lionel. "Numerical comparisons of two long-wave limit models." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.3 (2004): 419-436. <http://eudml.org/doc/245192>.

@article{Labbé2004,
abstract = {The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations 16 (2003) 1039–1064; Pego and Quintero, Physica D 132 (1999) 476–496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically the link between (KP) and (BL) and we point out the coupling effects emerging by considering two solitary waves propagating in two opposite directions.},
author = {Labbé, Stéphane, Paumond, Lionel},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Benney-Luke; Kadomtsev-Petviashvili; spectral method; long wave limit},
language = {eng},
number = {3},
pages = {419-436},
publisher = {EDP-Sciences},
title = {Numerical comparisons of two long-wave limit models},
url = {http://eudml.org/doc/245192},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Labbé, Stéphane
AU - Paumond, Lionel
TI - Numerical comparisons of two long-wave limit models
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 3
SP - 419
EP - 436
AB - The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations 16 (2003) 1039–1064; Pego and Quintero, Physica D 132 (1999) 476–496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically the link between (KP) and (BL) and we point out the coupling effects emerging by considering two solitary waves propagating in two opposite directions.
LA - eng
KW - Benney-Luke; Kadomtsev-Petviashvili; spectral method; long wave limit
UR - http://eudml.org/doc/245192
ER -

References

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  1. [1] M.J. Ablowitz and H. Segur, On the evolution of packets of water waves. J. Fluid Mech. 92 (1979) 691–715. Zbl0413.76009
  2. [2] J.C. Alexander, R.L. Pego and R.L. Sachs, On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation. Phys. Lett. A 226 (1997) 187–192. Zbl0962.35505
  3. [3] W. Ben Youssef and T. Colin, Rigorous derivation of Korteweg-de Vries-type systems from a general class of nonlinear hyperbolic systems. ESAIM: M2AN 34 (2000) 873–911. Zbl0962.35152
  4. [4] W. Ben Youssef and D. Lannes, The long wave limit for a general class of 2D quasilinear hyperbolic problems. Comm. Partial Differ. Equations 27 (2002) 979–1020. Zbl1072.35572
  5. [5] D.J. Benney and J.C. Luke, On the interactions of permanent waves of finite amplitude. J. Math. Phys. 43 (1964) 309–313. Zbl0128.44601
  6. [6] K.M. Berger and P.A. Milewski, The generation and evolution of lump solitary waves in surface-tension-dominated flows. SIAM J. Appl. Math. 61 (2002) 731–750 (electronic). Zbl1136.76325
  7. [7] J.L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves. Preprint Université de Bordeaux I, U-03-22 (2003). Zbl1050.76006MR2196497
  8. [8] W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Comm. Partial Differ. Equations 10 (1985) 787–1003. Zbl0577.76030
  9. [9] T. Gallay and G. Schneider, KP description of unidirectional long waves. The model case. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 885–898. Zbl1015.76015
  10. [10] T. Kano and T. Nishida, A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves. Osaka J. Math. 23 (1986) 389–413. Zbl0622.76021
  11. [11] D. Lannes, Consistency of the KP approximation. Discrete Contin. Dyn. Syst. (suppl.) (2003) 517–525. Dynam. Syst. Differ. equations (Wilmington, NC, 2002). Zbl1066.35017
  12. [12] P.A. Milewski and J.B. Keller, Three-dimensional water waves. Stud. Appl. Math. 97 (1996) 149–166. Zbl0860.35115
  13. [13] P.A. Milewski and E.G. Tabak, A pseudospectral procedure for the solution of nonlinear wave equations with examples from free-surface flows. SIAM J. Sci. Comput. 21 (1999) 1102–1114 (electronic). Zbl0953.65073
  14. [14] L. Paumond, Towards a rigorous derivation of the fifth order KP equation. Submitted for publication (2002). Zbl1137.35424
  15. [15] L. Paumond, A rigorous link between KP and a Benney-Luke equation. Differential Integral Equations 16 (2003) 1039–1064. Zbl1056.76014
  16. [16] R.L. Pego and J.R. Quintero, Two-dimensional solitary waves for a Benney-Luke equation. Physica D 132 (1999) 476–496. Zbl0935.35139
  17. [17] G. Schneider and C.E. Wayne, The long-wave limit for the water wave problem. I. The case of zero surface tension. Comm. Pure Appl. Math. 53 (2000) 1475–1535. Zbl1034.76011

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