Spectral Methods For Two Classes Of Parabolic Equations

As important ways for numerical solutions of partial differential equations, spectral methods have been developed rapidly in the past several decades. They not only have been widely applied to numerical simulations in various fields such as physics, mechanics, aerology, oceanics etc, and their theories of numerical analysis also tend to perfect. Owing to the less calculation of FFT and theirs so-called convergence of "infinite order", spectral methods have been paid more attention.In this paper, we deal with two classes of nonlinear PDEs using Fourier-Galerkin method. At first, we discuss the Benney equation with the periodic initial and boundary value problem. The semidiscrete scheme is constructed, the existence, uniqueness and stability are proved for the generalized solution. On the basis of semidiscrete scheme, we discretize time and construct the fully discrete scheme. The H~1— error estimates are given and the convergence is proved for the approximate solution of the spectral methods. Moreover, the propagation of solitary is simulated. Second, we discuss the evolution problem of sand ripples, the semidiscrete scheme is constructed and the H~2—error estimate is proved. Similarly, we also simulate the equation and get some results which are consistent with facts on the whole.