# American Institute of Mathematical Sciences

September  2020, 25(9): 3749-3763. doi: 10.3934/dcdsb.2020089

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

 †. School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu, Sichuan 611731, China ‡. Department of Mathematics, University of South Carolina Columbia, SC 29208

Received  May 2019 Revised  July 2019 Published  April 2020

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].

Citation: Zeyu Xia, Xiaofeng Yang. A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3749-3763. doi: 10.3934/dcdsb.2020089
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##### References:
Discrete $\ell^2$ and $\ell^\infty$ numerical errors for $D_N^2 u$ at $T = 4.0$, plotted versus $N$, the number of spatial grid point, for the numerical scheme (2.10). The time step size is fixed as ${\Delta t} = 10^{-4}$. An apparent spatial spectral accuracy is observed for both norms
Discrete $\ell^2$ and $\ell^\infty$ numerical errors for $D_N^2 u$ at $T = 4.0$, plotted versus $N_K$, the number of time steps, for the numerical scheme (2.10). The spatial resolution is fixed as $N = 512$. The data lie roughly on curves $CN_K^{-2}$, for appropriate choices of $C$, confirming the full second-order temporal accuracy of the scheme
The numerical solutions at a sequence of time instants. The outer solid line, the outer dashed line, the inner solid line and the inner dashed line stand for the numerical solutions $t = 4$, $8$, $12$ and $16$, respectively. The initial data is given by (5.3), and the physical domain is set as $\Omega = (-40, 40)$
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