High-order conservative schemes for the nonlinear Schrödinger equation in the semiclassical limit

Abstract

This letter devotes to the design of efficient prediction–correction numerical methods which produce high-order approximations of the solutions while preserving mass, or energy, or both of them, for the semiclassical Schrödinger equation with small Planck constant ɛ. The prediction step involves an explicit temporal fourth-order exponential Runge–Kutta method which allows the ɛ-oscillatory solution to be captured efficiently. The correction step only requires solving algebraic nonlinear equations. Numerical results show that the present methods have good meshing strategies τ=O(ɛ) and h=O(ɛ) and excellent power in the simulation of Bose–Einstein condensation.