A Boussinesq system for two-way propagation of nonlinear dispersive waves

Abstract

In this report, we study the system N t + W x + (NW) x − 1 6 N xxt = 0 , W t + N x + WW x − 1 6 W xxt = 0 , (∗) which describes approximately the two-dimensional propagation of surface waves in a uniform horizontal channel of length L 0 filled with an irrotational, incompressible, inviscid fluid which in its undisturbed state has depth h. The non-dimensional variables N( x, t) and W( x, t) represent the deviation of the water surface from its undisturbed position and the horizontal velocity at water level 2 3h , respectively. The natural initial-boundary-value problem corresponding to the situation wherein the channel is fitted with a wavemaker at both ends is formulated and analyzed theoretically. We then present a numerical algorithm for the approximation of solutions of the system (∗) and prove the algorithm is fourth-order accurate both in time and in space, is unconditionally stable, and has optimal computational complexity, which is to say the operation cost per time step is of order M where M is the number of grid points in the spatial discretization. Further, we implement the algorithm as a computer code and use it to study head-on collisions of solitary waves. Our numerical simulations are compared with existing theoretical, numerical and experimental results. We have tentatively concluded that the system (∗) is a good candidate for modeling two-dimensional surface water waves.