Efficient collocation methods for Volterra integral equations with highly oscillatory kernel

In this paper, we construct efficient collocation methods to deal with linear Volterra integral equations with highly oscillatory kernel. To discretize the oscillatory integrals in the collocation equation, a Filon type method is adopted. Based on some lemmas, we analyze the asymptotic property of the solution and obtain the convergence of the method, which is related to both the frequency ω and step length h. The proposed method does not only have the same order at the collocation points as the classical collocation method, but also may enjoy an asymptotic order which could reach two in some cases. Some numerical examples are presented at last to show the theoretical results and the efficiency of the method.