scholarly journals Quadrature methods for multivariate highly oscillatory integrals using derivatives

2006 ◽  
Vol 75 (255) ◽  
pp. 1233-1259 ◽  
Author(s):  
Arieh Iserles ◽  
Syvert P. Nørsett
Keyword(s):  
Oscillatory Integrals ◽  
Quadrature Methods ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals

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 Fast Computation of Integrals with Fourier-Type Oscillator Involving Stationary Point

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1160 ◽  
Author(s):  
Sakhi Zaman ◽  
Irshad Hussain ◽  
Dhananjay Singh
Keyword(s):  
Collocation Method ◽  
Stationary Point ◽  
Numerical Evaluation ◽  
Oscillatory Integrals ◽  
Test Problems ◽  
Fast Computation ◽  
Splitting Algorithm ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals ◽  
Fourier Type

An adaptive splitting algorithm was implemented for numerical evaluation of Fourier-type highly oscillatory integrals involving stationary point. Accordingly, a modified Levin collocation method was coupled with multi-resolution quadratures in order to tackle the stationary point and irregular oscillations of the integrand caused by ω . Some test problems are included to verify the accuracy of the proposed methods.

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