Multiple quadrature using highly oscillatory quadrature methods

2004 ◽  
Vol 163 (1) ◽  
pp. 1-13 ◽  
Author(s):  
G.A. Evans
Keyword(s):  
Quadrature Methods ◽  
Highly Oscillatory ◽  
Oscillatory Quadrature

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.

scholarly journals Superinterpolation in Highly Oscillatory Quadrature

2011 ◽  
Vol 12 (2) ◽  
pp. 203-228 ◽  
Author(s):  
Daan Huybrechs ◽  
Sheehan Olver
Keyword(s):  
Highly Oscillatory ◽  
Oscillatory Quadrature
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