  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.
