Moment-free numerical integration of highly oscillatory functions

Abstract

The aim of this paper is to derive new methods for numerically approximating the integral of a highly oscillatory function. We begin with a review of the asymptotic and Filon-type methods developed by Iserles and Nørsett. Using a method developed by Levin as a point of departure, we construct a new method that utilizes the same information as a Filon-type method, and obtains the same asymptotic order, while not requiring the computation of moments. We also show that a special case of this method has the property that the asymptotic order increases with the addition of sample points within the interval of integration, unlike all the preceding methods whose orders depend only on the endpoints.