Numerical comparisons of two long-wave limit models
ESAIM: Modélisation mathématique et analyse numérique, Volume 38 (2004) no. 3, pp. 419-436.

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.

DOI: 10.1051/m2an:2004020
Classification: 35L05, 35Q51, 35Q53, 65J15, 65M70
Keywords: Benney-Luke, Kadomtsev-Petviashvili, spectral method, long wave limit
     author = {Labb\'e, St\'ephane and Paumond, Lionel},
     title = {Numerical comparisons of two long-wave limit models},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {419--436},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {3},
     year = {2004},
     doi = {10.1051/m2an:2004020},
     mrnumber = {2075753},
     zbl = {1130.76324},
     language = {en},
     url = {}
AU  - Labbé, Stéphane
AU  - Paumond, Lionel
TI  - Numerical comparisons of two long-wave limit models
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2004
SP  - 419
EP  - 436
VL  - 38
IS  - 3
PB  - EDP-Sciences
UR  -
DO  - 10.1051/m2an:2004020
LA  - en
ID  - M2AN_2004__38_3_419_0
ER  - 
%0 Journal Article
%A Labbé, Stéphane
%A Paumond, Lionel
%T Numerical comparisons of two long-wave limit models
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2004
%P 419-436
%V 38
%N 3
%I EDP-Sciences
%R 10.1051/m2an:2004020
%G en
%F M2AN_2004__38_3_419_0
Labbé, Stéphane; Paumond, Lionel. Numerical comparisons of two long-wave limit models. ESAIM: Modélisation mathématique et analyse numérique, Volume 38 (2004) no. 3, pp. 419-436. doi : 10.1051/m2an:2004020.

[1] M.J. Ablowitz and H. Segur, On the evolution of packets of water waves. J. Fluid Mech. 92 (1979) 691-715. | Zbl

[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. | Zbl

[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. | Numdam | Zbl

[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. | Zbl

[5] D.J. Benney and J.C. Luke, On the interactions of permanent waves of finite amplitude. J. Math. Phys. 43 (1964) 309-313. | Zbl

[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). | Zbl

[7] J.L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves. Preprint Université de Bordeaux I, U-03-22 (2003). | MR | Zbl

[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. | Zbl

[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. | Zbl

[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. | Zbl

[11] D. Lannes, Consistency of the KP approximation. Discrete Contin. Dyn. Syst. (suppl.) (2003) 517-525. Dynam. Syst. Differ. equations (Wilmington, NC, 2002). | Zbl

[12] P.A. Milewski and J.B. Keller, Three-dimensional water waves. Stud. Appl. Math. 97 (1996) 149-166. | Zbl

[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). | Zbl

[14] L. Paumond, Towards a rigorous derivation of the fifth order KP equation. Submitted for publication (2002). | Zbl

[15] L. Paumond, A rigorous link between KP and a Benney-Luke equation. Differential Integral Equations 16 (2003) 1039-1064. | Zbl

[16] R.L. Pego and J.R. Quintero, Two-dimensional solitary waves for a Benney-Luke equation. Physica D 132 (1999) 476-496. | Zbl

[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. | Zbl

Cited by Sources: