2012/07/10 by Vı́ctor Domínguez, Ivan G. Graham, Dominguez, V. +3 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #42A15 #65D30 #65Y20 #74J20 #Electromagnetic Scattering and Analysis #FOS: Mathematics #Mathematical functions and polynomials #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1207.2283
openalex publication_date 2012/07/10 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
In this paper we propose and analyse composite Filon-Clenshaw-Curtis quadrature rules for integrals of the form Ik[a,b](f,g) := ∫ab f(x) exp(ikg(x)) \rd x , where k ≥ 0, f may have integrable singularities and g may have stationary points. Our composite rule is defined on a mesh with M subintervals and requires MN+1 evaluations of f. It satisfies an error estimate of the form CN k-r M-N-1 + r, where r is determined by the strength of any singularity in f and the order of any stationary points in g and CN is a constant which is independent of k and M, but depends on N. The regularity requirements on f and g are explicit in the error estimates. For fixed k, the rate of convergence of the rule as M → ∞ is the same as would be obtained if f was smooth. Moreover, the quadrature error decays at least as fast as k → ∞ as does the original integral Ik[a,b](f,g). For the case of nonlinear oscillators g, the algorithm requires the evaluation of g-1 at non-stationary points. Numerical results demonstrate the sharpness of the theory. An application to the implementation of boundary integral methods for the high-frequency Helmholtz equation is given.