Abstract: The paper is concerned with the Kolmogorov-Spieqel-Siveshinky equation with periodic initial value problems. The long-time behavior of the solutions for semi-discrete Fourier pseudo-spectral scheme is studied.The convergence and the existence of the global attractors for the discrete system is proposed. The full-discrete Fourier pseudo-spectral explicit scheme and its convergence is discussed. Finally, the credibility of the scheme is examined by numerical examples to show that the scheme is stable for a long time and the step-length is long. The orbits in the phase space of the solution are presented.

Key words: Kolmogorov-Spieqel-Siveshinky equation, Semi-discrete Fourier pseudo-spectral method, Full-discrete Fourier pseudo-spectral method, Convergence