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In this paper, we discuss the nonlinear stability and convergence of a fullydiscrete Fourier pseudospectral method coupled with a specially designed secondorder time-stepping for the numerical solution of the "good" Boussinesqequation. Our analysis improves the existing results presented in earlierliterature in two ways. First, an $l_\infty(0, T^*; H2)$ convergence for thesolution and $l_\infty(0, T^*; l_2)$ convergence for the time-derivative of thesolution are obtained in this paper, instead of the $l_\infty(0, T^*; l_2)$convergence for the solution and the $l_\infty(0, T^*; H^{-2})$ convergence forthe time-derivative, given in [17]. In addition, the stability and convergenceof this method is shown to be unconditional for the time step in terms of thespatial grid size, compared with a severe restriction time step restriction$\Delta t \leq Ch^2$ reported in [17].

A Fourier Pseudospectral Method for the "Good" Boussinesq Equation with Second Order Temporal Accuracy