Mathematical Models and Numerical Methods for High Frequency Waves

The numerical approximation of high frequency wave propagation is important in many applications. Examples include the simulation of seismic, acoustic, optical waves and microwaves. When the frequency of the waves is high, this is a difficult multiscale problem. The wavelength is short compared to the overall size of the computational domain and direct simulation using the standard wave equations is very expensive. Fortunately, there are computationally much less costly models, that are good approximations of many wave equations precisely for very high frequencies. Even for linear wave equations these models are often nonlinear. The goal of this paper is to review such mathematical models for high frequency waves, and to survey numerical methods used in simulations. We focus on the geometrical optics approximation which describes the infinite frequency limit of wave equations. We will also discuss finite frequency corrections and some other models. AMS subject classfications: 65-02, 35L05, 78A05