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].

中文翻译：

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时间的二阶精度，“ Good” Boussinesq方程的傅里叶伪谱数值格式

本文提供了“ Good” Boussinesq方程数值方案的非线性稳定性和收敛性，具有二阶时间精度和空间傅立叶拟谱近似。代替引入中间变量$ \ psi $来近似一阶时间导数，我们对二阶时间导数应用直接近似，这反过来导致中间数值变量的减少和计算效率的提高。仔细的分析揭示了这种时间离散化的无条件稳定性和收敛性。此外，通过使用别名错误控制技术，我们获得了$ u $和$ \ ell ^ \ infty（0,0，T ^ *; H ^ 2）$收敛。 T ^ *;19]。