Multiscale Methods and Analysis for Highly Oscillatory Differential Equations

Abstract

The oscillatory phenomena happen everywhere in our life, ranging from macroscopic to microscopic level. They are usually governed by some highly oscillatory nonlinear differential equations from either classical or quantum mechanics. Finding effective and accurate approximation to the highly oscillatory equations is the key way to study these nonlinear phenomena with oscillations in different scientific research fields. In this thesis, we propose and analyze some efficient 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 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.