# Numerical comparisons of two long-wave limit models

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m2an0364.dvi ESAIM: M2AN ESAIM: Mathematical Modelling and Numerical Analysis Vol. 38, No 3, 2004, pp. 419–436 DOI: 10.1051/m2an:2004020 NUMERICAL COMPARISONS OF TWO LONG-WAVE LIMIT MODELS Ste´phane Labbe´1 and Lionel Paumond1 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. Mathematics Subject Classification. 35L05, 35Q51, 35Q53, 65J15, 65M70. Received: October 3, 2003. Introduction The difficulties met to work with the full water-wave problem lead to derive simplified model in the special case of long waves. The model equation we will consider describe the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude, Φtt −∆Φ + µ(a∆2Φ− b∆Φtt) + ε(Φt∆Φ + (∇Φ)2t ) = 0, (1) where Φ(x, y, t) is the velocity potential on the bottom of the domain after rescaling the variables and ∇ and ∆ are respectively the two-dimensional gradient and Laplacian. The constants a and b are positive and such that a − b = σ − 13 �= 0 where σ is the Bond number. ε is the amplitude parameter (nonlinearity coefficient) and µ = (h0/L)2 is the long wave parameter (dispersion coefficient), where h0 is the equilibrium depth and L is the length scale. This equation was first derived by Benney and Luke (BL) (see [5]) when a = 1/6 and b = 1/2 with no surface tension. Let us remark that the model (1) is not valid for a = b (σ = 1/3) and in this case, it corresponds to a nonli

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