Approximate Conservation Laws for an Integrable Boussinesq System

The so-called Kaup-Boussinesq system is a model for long waves propagating at the surface of a perfect fluid. In this work, a derivation of approximate local conservation equations associated to the Kaup-Boussinesq system is given. The derivation of the approximate balance laws is based on reconstruction of the velocity field and the pressure in the fluid column below the free surface, and yields expressions for mass, momentum and energy densities and the corresponding fluxes. It is shown that the total energy found with this method is equal to the Hamiltonian functional featuring in the work of Craig and Groves [10]. For the numerical approximation of solutions to the Kaup-Boussinesq system, a filtered spectral method is put forward and shown to be stable when coupled with a convergent time -stepping scheme. The spectral method is used to confirm the exact conservation of the total momentum and energy.