Numerical solution of the ‘classical’ Boussinesq system

We consider the ‘classical’ Boussinesq system of water wave theory, which belongs to the class of Boussinesq systems modelling two‐way propagation of long waves of small amplitude on the surface of water in a horizontal channel. (We also consider its completely symmetric analog.) We discretize the initial‐boundary‐value problem for these systems, corresponding to homogeneous Dirichlet boundary conditions on the velocity variable at the endpoints of a finite interval, using fully discrete Galerkin‐finite element methods of high accuracy. We use the numerical schemes as exploratory tools to study the propagation and interactions of solitary‐wave solutions of these systems, as well as other properties of their solutions.