A time-discrete pseudospectral algorithm is given for the numerical solution of the nonlinear, periodic initial value problem where &bgr;, &ggr;, &egr;, and &dgr; are real constants greater than zero and q is 2&pgr;-periodic. For J ∈ &NN;, set h =&pgr;/J and consider x j = j h for j = 0 , ∓ 1 , ∓ 2 ,... Let U(t) be an approximate grid restriction of the solution u(.,t). For discretization in time, consider time levels t n = n k where n = 0 , 1 ,... and N = [ T&slash;k]. Let Un denote approximation to un, the grid restriction of u(.,t). The scheme presented for implementation is where - J ≤ p ≤ J and 1 ≤ n ≤ N - 1; Ypn and Zpn denote the pth Fourier coefficients of [Un]p^ and [(Un)2]p^ defined by [V]p^=( 1/2&pgr; )′′hVjexp(−ipjh) for - J ≤ p ≤ J (the double prime on the summation indicates that the first and last terms are halved).
The authors analyze the nonlinear stability and convergence of the scheme. For the truncation errors, under certain assumptions, they derive the following bound in terms of the energy norm introduced: where the parameter s, denoting the order of consistency in space, depends on the smoothness of u. Adding the assumptions that s>1/2, the starting vectors provide O ( k 2 + h s ) approximations, and grids are refined such that k = o ( h ¼ ) , h → 0, then A numerical example illustrates the superiority, in terms of accuracy and cost, of this pseudospectral method over a previously known method.
