Reproducing-Kernel Based Filon Method for Highly Oscillatory Integrals
DOI:
https://doi.org/10.48165/gjs.2026.3201Keywords:
Highly oscillatory integrals, Filon method, Reproducing-kernel, Runge's phenomenon, Asymptotic quadratureAbstract
Highly oscillatory integrals (HOIs) with Fourier-type kernels arise in electromagnetic scattering, wave optics, acoustics and quantum mechanics [26] and their accurate numerical evaluation remains difficult by classical quadrature rules once the frequency parameter becomes large. The classical Filon method resolves this difficulty by interpolating the slowly varying amplitude function with a polynomial basis and the resulting oscillatory moments can be evaluated analytically. However, monomial interpolation is well known to suffer from Runge's phenomenon at higher numbers of nodes which degrades accuracy of the Filon method. In this paper, a reproducing-kernel based Filon method is developed in which the amplitude function is represented in a reproducing-kernel Hilbert space (RKHS) instead of the monomial basis which suppresses Runge-type oscillations. The resulting moments are evaluated by asymptotic expansion. Alternatively, Taylor polynomial is used to interpolate the amplitude function. For integrands with an interior or boundary stationary point, a splitting technique combined with a hybrid-function quadrature is used to obtain a Filon-hybrid scheme. Several benchmark problems including Runge-type amplitude function are considered for evaluation to validate the proposed schemes. The numerical results confirm that the reproducing-kernel based Filon method attains markedly higher accuracy and better convergence rate than both the monomial based Filon method and multiquadric RBF-based collocation.
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