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

Abstract

In this paper, we discuss the nonlinear stability and convergence of a fully discrete Fourier pseudospectral method coupled with a specially designed second order time-stepping for the numerical solution of the "good" Boussinesq equation. Our analysis improves the existing results presented in earlier literature in two ways. First, an $l_{\infty }(0,T^{\ast };H2)$ convergence for the solution and $l_{\infty }(0,T^{\ast };l_2)$ convergence for the time-derivative of the solution are obtained in this paper, instead of the $l_{\infty }(0,T^{\ast };l_2)$ convergence for the solution and the $l_{\infty }(0,T^{\ast };H^{-2})$ convergence for the time-derivative, given in [17]. In addition, the stability and convergence of this method is shown to be unconditional for the time step in terms of the spatial grid size, compared with a severe restriction time step restriction $\Delta t\le C{h}^2$ reported in [17].

Cited by in corpus (1)

• An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation