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.

翻译：本文介绍了用于数值求解一类色散演化方程的滤波有限差分法，该方程的解在空间和时间上均具有高度振荡特性。我们考虑一个具有半经典尺度的非线性薛定谔方程，其初始数据为调制的平面波形式，具有高度振荡性。所提出的方法无需像标准有限差分法那样，通过计算上不可行的精细网格来分辨空间和时间上的高频振荡。本文采用的方法通过引入适当的滤波器来改进传统的有限差分格式。具体而言，我们提出了滤波蛙跳法和滤波Crank--Nicolson法，这两种方法均能达到二阶精度，且其时间步长和网格尺寸不受小半经典参数量级的限制。此外，滤波Crank--Nicolson法能够同时保持离散质量和离散能量的守恒性。数值实验验证了理论结果。