A Symplectic Numerical Method for Boussinesq Equation

In this paper we study the fourth order Boussinesq equation with a single nonlinearity. We rewrite this equation in a Hamiltonian form and present a noncompact finite difference scheme, which preserves the symplectic structure of the initial problem. We construct a new finite difference scheme, which is very similar to the symplectic scheme but preserves the discrete energy exactly. To illustrate both schemes we present many computational results. We compare the performance of the symplectic scheme with the new conservative finite difference scheme.