SOLVING THE LONG-TIME NONLINEAR SCHRÖDINGER EQUATION BY A CLASS OF OSCILLATION-RELAXATION INTEGRATORS^*

In this paper, we numerically solve the long-time nonlinear Schrödinger equation (NLSE) by using an oscillation-relaxation formulation. Such a formulation allows us to propose a class of numerical methods named oscillation-relaxation integrators (ORIs) for NLSE. By convergence analysis, the pth order ORIs (for any p ≥ 1) are shown to offer error bounds of form ο(ε^2ph^p) up to time interval ο(ε^-2) with h > 0 the time stepsize. Compared with the splitting schemes that give error bounds ο(ε²h^p), the accuracy of ORIs are significantly improved. Numerical experiments and comparisons confirm the gain in practical efficiency. Long-term near-conservation laws of symmetric ORIs are also established and tested. Numerical exploration on longer time intervals show that ORIs can produce accurate results till times t ≫ ε^-2, thanks to the improved error constants.