IMPROVED UNIFORM ERROR BOUNDS OF THE TIME-SPLITTING METHODS FOR THE LONG-TIME (NONLINEAR) SCHRODINGER EQUATION

We establish improved uniform error bounds for the time-splitting methods for the long-time dynamics of the Schrodinger equation with small potential and the nonlinear Schrodinger equation (NLSE) with weak nonlinearity. For the Schrodinger equation with small potential characterized by a dimensionless parameter epsilon is an element of (0, 1], we employ the unitary flow property of the (second-order) time-splitting Fourier pseudospectral (TSFP) method in L-2-norm to prove a uniform error bound at time t(epsilon) = t/epsilon as C(t) C(T)(h(m) + tau(2)) up to t epsilon <= T epsilon - T/epsilon for any T > 0 and uniformly for epsilon is an element of (0, 1], while h is the mesh size, tau is the time step, m >= 2 and C(T) (the local error bound) depend on the regularity of the exact solution, and C(t) = C-0 + C(1)t grows at most linearly with respect to t with C-0 and C1 two positive constants independent of T, epsilon, h and tau. Then by introducing a new technique of regularity compensation oscillation (RCO) in which the high frequency modes are controlled by regularity and the low frequency modes are analyzed by phase cancellation and energy method, an improved uniform (w.r.t e) error bound at O(h(m- 1) + epsilon tau(2)) is established in H-1-norm for the long-time dynamics up to the time at O(1/epsilon) of the Schrodinger equation with O(epsilon)-potential with m >= 3. Moreover, the RCO technique is extended to prove an improved uniform error bound at O( h(m-1) + epsilon(2)tau(2)) in H-1- norm for the long-time dynamics up to the time at O( 1/epsilon(2)) of the cubic NLSE with O(epsilon(2))- nonlinearity strength. Extensions to the first-order and fourth-order time-splitting methods are discussed. Numerical results are reported to validate our error estimates and to demonstrate that they are sharp.