标题： 带有滤波的有限差分方法求解高度振荡的非线性 Klein–Gordon 方程 显示英文标题

标题： A filtered finite difference method for a highly oscillatory nonlinear Klein--Gordon equation

摘要： 我们考虑了一类非线性 Klein-Gordon 方程，在非相对论极限下，初始数据具有高度振荡的平面波形式。在此极限下，解在时间和空间上均表现出快速振荡，这对数值逼近提出了挑战。我们提出了一种滤波有限差分方法，该方法在时间步长和网格尺寸上不受到小参数的限制即可达到二阶精度。此外，该方法在从任意小到适度受限的尺度参数范围内具有统一收敛性。数值实验验证了理论结果。 显示英文摘要

摘要： We consider a nonlinear Klein--Gordon equation in the nonrelativistic limit regime with highly oscillatory initial data in the form of a modulated plane wave. In this regime, the solution exhibits rapid oscillations in both time and space, posing challenges for numerical approximation. We propose a filtered finite difference method that achieves second-order accuracy with time steps and mesh sizes that are not restricted in magnitude by the small parameter. Moreover, the method is uniformly convergent in the range from arbitrarily small to moderately bounded scaling parameters. Numerical experiments illustrate the theoretical results.