Filon quadrature

In numerical analysis, Filon quadrature or Filon's method is a technique for numerical integration of oscillatory integrals. It is named after English mathematician Louis Napoleon George Filon, who first described the method in 1934.[1]

Description

The method is applied to oscillatory definite integrals in the form:

${\displaystyle \int _{a}^{b}f(x)g(x)dx}$

where ${\textstyle f(x)}$ is a relatively slowly-varying function and ${\textstyle g(x)}$ is either sine or cosine or a complex exponential that causes the rapid oscillation of the integrand, particularly for high frequencies. In Filon quadrature, the ${\textstyle f(x)}$ is divided into ${\textstyle 2N}$ subintervals of length ${\textstyle h}$, which are then interpolated by parabolas. Since each subinterval is now converted into a Fourier integral of quadratic polynomials, these can be evaluated in closed-form by integration by parts. For the case of ${\textstyle g(x)=\cos(kx)}$, the integration formula is given as:[1][2]

${\displaystyle \int _{a}^{b}f(x)\cos(kx)dx\approx h(\alpha \left[f(b)\sin(kb)-f(a)\sin(ka)\right]+\beta C_{2n}+\gamma C_{2n-1})}$

where

${\displaystyle \alpha =\left(\theta ^{2}+\theta \sin(\theta )\cos(\theta )-2\sin ^{2}(\theta )\right)/\theta ^{3}}$

${\displaystyle \beta =2\left[\theta (1+\cos ^{2}(\theta ))-2\sin(\theta )\cos(\theta )\right]/\theta ^{3}}$

${\displaystyle \gamma =4(\sin(\theta )-\theta \cos(\theta ))/\theta ^{3}}$

${\displaystyle C_{2n}={\frac {1}{2}}f(a)\cos(ka)+f(a+2h)\cos(k(a+2h))+f(a+4h)\cos(k(a+4h))+\ldots +{\frac {1}{2}}f(b)\cos(kb)}$

${\displaystyle C_{2n-1}=f(a+h)\cos(k(a+h))+f(a+3h)\cos(k(a+3h))+\ldots +f(b-h)\cos(k(b-h))}$

${\displaystyle \theta =kh}$

Explicit Filon integration formulas for sine and complex exponential functions can be derived similarly.[2] The formulas above fail for small ${\textstyle \theta }$ values due to catastrophic cancellation;[3] Taylor series approximations must be in such cases to mitigate numerical errors, with ${\textstyle \theta =1/6}$ being recommended as a possible switchover point for 44-bit mantissa.[2]

Modifications, extensions and generalizations of Filon quadrature have been reported in numerical analysis and applied mathematics literature; these are known as Filon-type integration methods.[4][5] These include Filon-trapezoidal[2] and Filon–Clenshaw–Curtis methods.[6]

Applications

Filon quadrature is widely used in physics and engineering for robust computation of Fourier-type integrals. Applications include evaluation of oscillatory Sommerfeld integrals for electromagnetic and seismic problems in layered media[7][8][9] and numerical solution to steady incompressible flow problems in fluid mechanics,[10] as well as various different problems in neutron scattering,[11] quantum mechanics[12] and metallurgy.[13]

See also

• Tanh-sinh quadrature

References

1. Filon, L. N. G. (1930). "III.—On a Quadrature Formula for Trigonometric Integrals". Proceedings of the Royal Society of Edinburgh. 49: 38–47. doi:10.1017/S0370164600026262.
2. Davis, Philip J.; Rabinowitz, Philip (1984). Methods of Numerical Integration (2 ed.). Academic Press. pp. 151–160. ISBN 9781483264288.
3. Chase, Stephen M.; Fosdick, Lloyd D. (1969). "An algorithm for Filon quadrature". Communications of the ACM. 12 (8): 453–457. doi:10.1145/363196.363209.
4. Iserles, A.; Nørsett, S. P. (2004). "On quadrature methods for highly oscillatory integrals and their implementation". BIT Numerical Mathematics. 44 (4): 755–772. doi:10.1007/s10543-004-5243-3.
5. Xiang, Shuhuang (2007). "Efficient Filon-type methods for". Numerische Mathematik. 105: 633–658. doi:10.1007/s00211-006-0051-0.
6. Domínguez, V.; Graham, I. G.; Smyshlyaev, V. P. (2011). "Stability and error estimates for Filon–Clenshaw–Curtis rules for highly oscillatory integrals". IMA Journal of Numerical Analysis. 31 (4): 1253–1280. doi:10.1093/imanum/drq036.
7. Červený, Vlastislav; Ravindra, Ravi (1971). Theory of Seismic Head Waves. University of Toronto Press. pp. 287–289. ISBN 9780802000491.
8. Mosig, J. R.; Gardiol, F. E. (1983). "Analytical and numerical techniques in the Green's function treatment of microstrip antennas and scatterers". IEE Proceedings H - Microwaves, Optics and Antennas. 130 (2): 175–182. doi:10.1049/ip-h-1.1983.0029.
9. Chew, Weng Cho (1990). Waves and Fields in Inhomogeneous Media. New York: Van Nostrand Reinhold. p. 118. ISBN 9780780347496.
10. Dennis, S. C. R.; Chang, Gau-Zu (1970). "Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100". Journal of Fluid Mechanics. 42 (3): 471–489. Bibcode:1970JFM....42..471D. doi:10.1017/S0022112070001428.
11. Grimley, David I.; Wright, Adrian C.; Sinclair, Roger N. (1990). "Neutron scattering from vitreous silica IV. Time-of-flight diffraction". Journal of Non-Crystalline Solids. 119 (1): 49–64. Bibcode:1990JNCS..119...49G. doi:10.1016/0022-3093(90)90240-M.
12. Fedotov, A.; Ilderton, A.; Karbstein, F.; King, B.; Seipt, D.; Taya, H.; Torgrimsson, G. (2023). "Advances in QED with intense background fields". Physics Reports. 1010: 1–138. arXiv:2203.00019. Bibcode:2023PhR..1010....1F. doi:10.1016/j.physrep.2023.01.003.
13. Thouless, M. D.; Evans, A. G.; Ashby, M. F.; Hutchinson, J. W. (1987). "The edge cracking and spalling of brittle plates". Acta Metallurgica. 35 (6): 1333–1341. doi:10.1016/0001-6160(87)90015-0.