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]
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