Numerical comparisons of two long-wave limit models

Stéphane Labbé; Lionel Paumond

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • 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 Equations16 (2003) 1039–1064; Pego and Quintero, Physica D132 (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 38.3 (2010): 419-436. <http://eudml.org/doc/194221>.

@article{Labbé2010,
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 Equations16 (2003) 1039–1064; Pego and Quintero, Physica D132 (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},
keywords = {Benney-Luke; Kadomtsev-Petviashvili; spectral method; long wave limit.},
language = {eng},
month = {3},
number = {3},
pages = {419-436},
publisher = {EDP Sciences},
title = {Numerical comparisons of two long-wave limit models},
url = {http://eudml.org/doc/194221},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Labbé, Stéphane
AU - Paumond, Lionel
TI - Numerical comparisons of two long-wave limit models
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
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 Equations16 (2003) 1039–1064; Pego and Quintero, Physica D132 (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/194221
ER -

References

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  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. P.A. Milewski and J.B. Keller, Three-dimensional water waves. Stud. Appl. Math.97 (1996) 149–166.  Zbl0860.35115
  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. L. Paumond, Towards a rigorous derivation of the fifth order KP equation. Submitted for publication (2002).  Zbl1137.35424
  15. L. Paumond, A rigorous link between KP and a Benney-Luke equation. Differential Integral Equations16 (2003) 1039–1064.  Zbl1056.76014
  16. R.L. Pego and J.R. Quintero, Two-dimensional solitary waves for a Benney-Luke equation. Physica D132 (1999) 476–496.  Zbl0935.35139
  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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