On the Method of Neumann Series for Highly Oscillatory Equations

The main purpose of this paper is to describe and analyse techniques for the numerical solution of highily oscillatory ordinary differential equations by exploying a Neumann expansion. Once the variables in the differential system are changed with respect to a rapidly rotating frame of reference, the Neumann method becomes very effective indeed. However, this effectiveness rests upon suitable quadrature of highly oscillatory multivariate integrals, and we devote part of this paper to describe how to accomplish this to high accuracy with a modest computational effort.