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

Abstract

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 ψ 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 ∞ (0, T * ; H 2 ) convergence for u and ∞ (0, T * ; 2 ) convergence for the discrete time-derivative of the solution in this paper, in comparison with the ∞ (0, T * ; 2 ) convergence for u and the ∞ (0, T * ; H −2 ) convergence for the time-derivative, given in [19].

Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems

“…The estimates of I1+I2, III1+ III2, IV1+IV2 are identical to those for the proof of (18). To estimate II1+II2, we can either follow the inequality (35) exactly, or we can perform:…”

“…, by using the estimates (35) and (36), respectively. Using the conditions (31) and (32), respectively, leads to the desired inequality for the energy stability.…”

“…Note that an aliasing error will appear in the pseudo-spectral approximation. An aliasing error control technique was developed recently and applied for the 3-D viscous Burgers' equation [14], 3-D incompressible Navier-Stokes equation [33] and 'Good' Boussinesq equation [35]. We will take this technique into our consideration in the future work.…”