  In this paper, new Levin methods are presented for calculating oscillatory
integrals with algebraic and/or logarithmic singularities. To avoid
singularity, the technique of singularity separation is applied and then the
singular ODE occurring in classic Levin methods is converted into two kinds of
non-singular ODEs. The solutions of one can be obtained explicitly, while those
of the other can be solved efficiently by collocation methods. The proposed
methods can attach arbitrarily high asymptotic orders and also enjoy
superalgebraic convergence with respect to the number of collocation points.
Several numerical experiments are presented to validate the efficiency of the
proposed methods.
