Efficient quadrature of highly oscillatory integrals using derivatives

Author:

Iserles Arieh1,Nørsett Syvert P2

Affiliation:

1. Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK

2. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway

Abstract

In this paper, we explore quadrature methods for highly oscillatory integrals. Generalizing the method of stationary phase, we expand such integrals into asymptotic series in inverse powers of the frequency. The outcome is two families of methods, one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of Filon (Filon 1928 Proc. R. Soc. Edinb. 49 , 38–47) . Both kinds of methods approximate the integral as a linear combination of function values and derivatives, with coefficients that may depend on frequency. We determine asymptotic properties of these methods, proving, perhaps counterintuitively, that their performance drastically improves as frequency grows. The paper is accompanied by numerical results that demonstrate the potential of this set of ideas.

Publisher

The Royal Society

Subject

General Physics and Astronomy,General Engineering,General Mathematics

Reference12 articles.

1. Abramowitz M& Stegun I.A Handbook of mathematical functions. 1964 Washington DC:National Bureau of Standards.

2. Davis P.J& Rabinowitz P Methods of numerical integration. 2nd edn. 1980 Orlando FL:Academic Press.

3. Degani I. & Schiff J. 2003 RCMS: right correction Magnus series approach for integration of linear ordinary differential equations with highly oscillatory solution. Report Weizmann Institute of Science Israel.

4. III.—On a Quadrature Formula for Trigonometric Integrals

5. A Modification of Filon's Method of Numerical Integration

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