scholarly journals Moment-free numerical approximation of highly oscillatory integrals with stationary points

2007 ◽  
Vol 18 (4) ◽  
pp. 435-447 ◽  
Author(s):  
SHEEHAN OLVER
Keyword(s):  
Asymptotic Expansion ◽  
Stationary Phase ◽  
Asymptotic Behaviour ◽  
Approximation Method ◽  
Numerical Approximation ◽  
Oscillatory Integrals ◽  
Numerical Quadrature ◽  
Stationary Points ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals

This article presents a method for the numerical quadrature of highly oscillatory integrals with stationary points. We begin with the derivation of a new asymptotic expansion, which has the property that the accuracy improves as the frequency of oscillations increases. This asymptotic expansion is closely related to the method of stationary phase, but presented in a way that allows the derivation of an alternate approximation method that has similar asymptotic behaviour, but with significantly greater accuracy. This approximation method does not require moments.

scholarly journals Efficient quadrature of highly oscillatory integrals using derivatives

2005 ◽  
Vol 461 (2057) ◽  
pp. 1383-1399 ◽  
Author(s):  
Arieh Iserles ◽  
Syvert P Nørsett
Keyword(s):  
Stationary Phase ◽  
Linear Combination ◽  
Asymptotic Properties ◽  
Asymptotic Series ◽  
Numerical Results ◽  
Oscillatory Integrals ◽  
The Other ◽  
Quadrature Methods ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals

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.

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