A combination of multiscale time integrator and two-scale formulation for the nonlinear Schrödinger equation with wave operator

In this paper, we consider the nonlinear Schrödinger equation with wave operator (NLSW), which contains a dimensionless parameter 0<ε≤1. As 0<ε≪1, the solution of the NLSW propagates fast waves in time with wavelength O(ε2) and the problem becomes highly oscillatory in time. The oscillations come from two parts. One part is from the equation and another part is from the initial data. For the ill-prepared initial data case as described in Bao and Cai (2014) which brings inconsistency in the limit regime, standard numerical methods have strong convergence order reduction in time when ε becomes small. We review two existing methods to solve the NLSW: an exponential integrator and a two-scale method. We comment on their order reduction issues. Then we derive a multiscale decomposition two-scale method for solving the NLSW by first performing a multiscale decomposition on the NLSW which decomposes it into a well-behaved part and an energy-unbounded part, and then applying an exponential integrator for the well-behaved part and a two-scale approach for the energy-unbounded part. Numerical experiments are conducted to test the proposed method which shows uniform second order accuracy without significant order reduction for all 0<ε≤1. Comparisons are made with the existing methods.