Weighted finite difference methods for the semiclassical nonlinear Schr\"odinger equation with multiphase oscillatory initial data

Abstract

This paper introduces weighted finite difference methods for numerically solving dispersive evolution equations with solutions that are highly oscillatory in both space and time. We consider a semiclassically scaled cubic nonlinear Schr\"odinger equation with highly oscillatory initial data, first in the single-phase case and then in the general multiphase case. The proposed methods do not need to resolve high-frequency oscillations in both space and time by prohibitively fine grids as would be required by standard finite difference methods. The approach taken here modifies traditional finite difference methods by appropriate exponential weights. Specifically, we propose the weighted leapfrog and weighted Crank--Nicolson methods, both of which achieve second-order accuracy with time steps and mesh sizes that are not restricted in magnitude by the small semiclassical parameter. Numerical experiments illustrate the theoretical results.