High frequency quadrature and its application in scattering problems

Highly oscillating integrals have drawn interest since the early stages of real and complex analysis. Their role in scattering theory, optics, fluid dynamics and many other applications increased this interest, and in the early 2000s study of numerical computation of such integrals leaped forward. Several methods are proposed for numerical methods, which we briefly compare in the thesis. In this thesis we survey the Filon-Clenshaw-Curtis quadrature for highly oscillating integrals with algebraic and logarithmic singularities and oscillators with stationary points. In that matter, several algorithms are developed and error analysis is presented. The algorithms utilize the advantageous properties of Filon-Clenshaw-Curtis quadrature compared to previously proposed numerical methods. Numerical experiments which investigate extreme cases show the accuracy and stability of the proposed quadrature rule.