**Hao Chen, Ling Liu and Junjie Ma \***

School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China; sdch0807@163.com (H.C.); lingliu95@126.com (L.L.)

**\*** Correspondence: jjma@gzu.edu.cn

Received: 9 October 2020; Accepted: 4 November 2020; Published: 10 November 2020

**Abstract:** In this work, we introduce a class of generalized multistep collocation methods for solving oscillatory Volterra integral equations, and study two kinds of convergence analysis. The error estimate with respect to the stepsize is given based on the interpolation remainder, and the nonclassical convergence analysis with respect to oscillation is developed by investigating the asymptotic property of highly oscillatory integrals. Besides, the linear stability is analyzed with the help of generalized Schur polynomials. Several numerical tests are given to show that the numerical results coincide with our theoretical estimates.

**Keywords:** collocation; volterra integral equation; highly oscillatory; convergence
