Integration of highly oscillatory functions

Abstract

Integrals of slowly decaying oscillating functions over unbounded intervals can be computed by a combination of subdivision and extrapolation of the sequence of partial sums. This work presents guidelines how to choose an appropriate sequence of partition points based on a simple analysis of the oscillating factor of the integrand. It appears that the first subdivision point must be chosen with particular care in order to obtain correct extrapolation results. With the aid of numerical examples, we investigate several integration methods suitable for the actual evaluation of the partial integrals. It will be shown that for the two distinct types of integrals involved, two different methods should be employed to obtain best performance. An example of wave propagation through a dispersive medium finally illustrates the practicability of the approach, which can be implemented using only standard numerical algorithms.