Augmented Levin methods for vector-valued highly oscillatory integrals with exotic oscillators and turning points

Abstract

In this paper, we propose an efficient Levin method to approximate vector-valued highly oscillatory integrals with exotic oscillators and turning points. These include integrals involving Hankel functions, the product of Hankel functions with exponential functions, or the product of different Bessel functions. The problem of Levin methods encountering turning points remains an open problem (S. Olver, BIT Numerical Mathematics, 47(3):637-655, 2007). To address the difficulties caused by the turning points, the original Levin ordinary differential equation (Levin-ODE) is converted into augmented ordinary differential equations based on the superposition theory, which can be solved efficiently by the spectral collocation method together with Meijer G-functions. Four kinds of vector-valued highly oscillatory integrals are considered and numerical examples are presented to show the effectiveness and accuracy of the proposed methods.