A bivariate Filon–Clenshaw–Curtis method of the highly oscillatory integrals on a square

Abstract

We present a generalised bivariate Filon–Clenshaw–Curtis cubature for the double highly oscillatory integrals on the square. Under the Hermite interpolatory conditions meeting an arbitrary asymptotic order, one unisolvent Hermite interpolatory interpolation is introduced which contributes to the error of Filon method. The basic properties and error of the proposed method are analysed under the chosen interpolator space. With the help of the discrete cosine transform, the generalised bivariate Filon–Clenshaw–Curtis method is constructed which can achieve a good accuracy for the whole oscillator parameters with less computational complexity. Numerical experiments are provided to illustrate the effectiveness of the method.