A second order operator splitting numerical scheme for the “good” Boussinesq equation

The nonlinear stability and convergence analyses are presented for a second order operator splitting scheme applied to the “good” Boussinesq equation, coupled with the Fourier pseudo-spectral approximation in space. Due to the wave equation nature of the model, we have to rewrite it as a system of two equations, for the original variable u and v=ut, respectively. In turn, the second order operator splitting method could be efficiently designed. A careful Taylor expansion indicates the second order truncation error of such a splitting approximation, and a linearized stability analysis for the numerical error function yields the desired convergence estimate in the energy norm. In more details, the convergence in the energy norm leads to an ℓ∞(0,T⁎;H2) convergence for the numerical solution u and ℓ∞(0,T⁎;ℓ2) convergence for v=ut. And also, the presented convergence is unconditional for the time step in terms of the spatial grid size, in comparison with a severe time step restriction, Δt≤Ch2, required in many existing works.