Filtered finite difference methods for nonlinear Schr\"odinger equations in semiclassical scaling

TL;DR

This paper develops filtered finite difference methods for the nonlinear Schrödinger equation in semiclassical scaling, enabling efficient computation without resolving high-frequency oscillations, and demonstrates their accuracy and conservation properties.

Contribution

It introduces filtered leapfrog and Crank--Nicolson methods that are second-order accurate and do not require fine grids dictated by the semiclassical parameter.

Findings

Methods achieve second-order accuracy.

Filtered methods do not need to resolve high-frequency oscillations.

The filtered Crank--Nicolson conserves discrete mass and energy.

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.