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]
The method is applied to oscillatory definite integrals in the form:
where is a relatively slowly-varying function and 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 is divided into subintervals of length , 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 , the integration formula is given as:[1][2]
where
Explicit Filon integration formulas for sine and complex exponential functions can be derived similarly.[2] The formulas above fail for small values due to catastrophic cancellation;[3] Taylor series approximations must be in such cases to mitigate numerical errors, with 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]
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]