Filtered finite difference methods for nonlinear Schr\"odinger equations in semiclassical scaling
Abstract
This paper introduces filtered finite difference methods for numerically solving a dispersive evolution equation with solutions that are highly oscillatory in both space and time. We consider a semiclassically scaled nonlinear Schr\"odinger equation with highly oscillatory initial data in the form of a modulated plane wave. 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 incorporating appropriate filters. Specifically, we propose the filtered leapfrog and filtered 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. Furthermore, the filtered Crank--Nicolson method conserves both the discrete mass and a discrete energy. Numerical experiments illustrate the theoretical results.
Cite
@article{arxiv.2411.07855,
title = {Filtered finite difference methods for nonlinear Schr\"odinger equations in semiclassical scaling},
author = {Yanyan Shi and Christian Lubich},
journal= {arXiv preprint arXiv:2411.07855},
year = {2025}
}
Comments
We have developed a new class of methods that significantly improve upon this version in addressing the semiclassical nonlinear Schr\"odinger equations, particularly in handling multiphase problems. The new approach differs substantially in both methodology and the generality of the problems considered. We will replace this submission with an updated version that reflects these improvements