A FIRST-ORDER FOURIER INTEGRATOR FOR THE NONLINEAR SCHRODINGER EQUATION ON T WITHOUT LOSS OF REGULARITY

In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schrodinger equation in one dimension. The scheme is explicit and can be implemented using the fast Fourier transform. By a rigorous analysis, we prove that the new scheme provides the first-order accuracy in H-gamma for any initial data belonging to H-gamma, for any gamma > 3/2. That is, up to some fixed time T, there exists some constant C = C(parallel to u parallel to L-infinity([0, T];H-gamma)) > 0, such that parallel to u(n) - u(t(n))parallel to H-gamma(T) <= C tau, where u(n) denotes the numerical solution at t(n) = n tau. Moreover, the mass of the numerical solution M(u(n)) verifies vertical bar M(u(n)) - M(u(0))| <= C tau(5). In particular, our scheme does not cost any additional derivative for the first-order convergence and the numerical solution obeys the almost mass conservation law. Furthermore, if u(0) is an element of H-1(T), we rigorously prove that parallel to u(n) - u(t(n))parallel to (H1(T)) <= C tau(1/2)-, where C = C(parallel to u(0)parallel to(H1(T))) > 0.