A Levin method for logarithmically singular oscillatory integrals

We propose a new stable Levin method to compute oscillatory integrals with logarithmic singularities and without stationary points. To avoid the singularity, we apply the technique of singularity separation and transform the singular ODE into two non-singular ODEs, which can be solved efficiently by the collocation method. Applying the equivalency of the new Levin method for the singular oscillatory integrals and the Filon method when the oscillator is linear, we consider the convergence of the new Levin method. This new method shares the proposition that less error for higher oscillation. Several numerical experiments are presented to validate the efficiency of the proposed method.