Abstract:

The oscillatory phenomena happen almost everywhere in our life, ranging from macroscopic to microscopic level. They are usually described and governed by some highly oscillatory nonlinear differential equations from either classical mechanics or quantum mechanics. Effective and accurate approximations to the highly oscillatory equations become the key way of further studies of the nonlinear phenomena with oscillations in different scientific research fields. In this talk, we propose and analyze some efficient numerical methods for approximating a class of highly oscillatory differential equations arising from quantum or plasma physics. The methods here include classical numerical discretizations and the multiscale methods with numerical implementations. Special attentions are paid to study the error bound of each numerical method in the highly oscillatory regime, which are geared to understand how the step size should be chosen in order to resolve the oscillations, and eventually to find out the uniformly accurate methods that could totally ignore the oscillations when approximating the equations.