A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation

Abstract

Listen

Translate article

The nonlinear stability and convergence of a numerical scheme for the 'Good' Boussinesq equation is provided in this article, with second order temporal accuracy and Fourier pseudo-spectral approximation in space. Instead of introducing an intermediate variable $ \psi $ to approximate the first order temporal derivative, we apply a direct approximation to the second order temporal derivative, which in turn leads to a reduction of the intermediate numerical variable and improvement in computational efficiency. A careful analysis reveals an unconditional stability and convergence for such a temporal discretization. In addition, by making use of the techniques of aliasing error control, we obtain an $ \ell^\infty (0,T^*; H^2) $ convergence for $ u $ and $ \ell^\infty (0,T^*; \ell^2) $ convergence for the discrete time-derivative of the solution in this paper, in comparison with the $ \ell^\infty (0,T^*; \ell^2) $ convergence for $ u $ and the $ \ell^\infty (0,T^*; H^{-2}) $ convergence for the time-derivative, given in [19].