On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation

On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation

The main theme of this paper is the construction of efficient, reliable and affordable error bounds for two families of quadrature methods for highly oscillatory integrals. We demonstrate, using asymptotic expansions, that the error can be bounded very precisely indeed at the cost of few extra derivative evaluations. Moreover, in place of derivatives it is possible to use finite difference appr...

On quadrature methods for highly oscillatory integrals and their implementations

Quadrature methods for highly oscillatory singular integrals

We study asymptotic expansions, Filon-type methods and complex-valued Gaussian quadrature for highly oscillatory integrals with power-law and logarithmic singularities. We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand, the stationary points and the endpoints of the integral. A truncated asymptotic expansion...

Quadrature methods for multivariate highly oscillatory integrals using derivatives

While there exist effective methods for univariate highly oscillatory quadrature, this is not the case in a multivariate setting. In this paper we embark on a project, extending univariate theory to more variables. Inter alia, we demonstrate that, subject to a nonresonance condition, an integral over a simplex can be expanded asymptotically using only function values and derivatives at the vert...

Asymptotic expansion and quadrature of composite highly oscillatory integrals

We consider in this paper asymptotic and numerical aspects of highly oscillatory integrals of the form ∫ b a f(x)g(sin[ωθ(x)])dx, where ω 1. Such integrals occur in the simulation of electronic circuits, but they are also of independent mathematical interest. The integral is expanded in asymptotic series in inverse powers of ω. This expansion clarifies the behaviour for large ω and also provide...