**1. Introduction**

In many practical problems, such as epidemic diffusion, population dynamics and reaction processes, one may usually come across a class of Volterra integral equations (VIEs) (see [1] and references therein). Noting that most VIEs cannot be solved in closed forms, many researchers have made contributions to the numerical approaches to VIEs.

Particularly, the study of numerical solutions to VIEs with highly oscillatory Fourier or Bessel kernels has attracted much attention during the past decade. In [2], Xiang and Brunner first investigated Filon collocation approximations to highly oscillatory VIEs by employing the asymptotic property of oscillatory integrals. They found that errors of Filon collocation solutions decayed fast as the frequency increased. The third author presented an optimal convergence order for the direct Filon collocation solution to the first kind of oscillatory VIE arising in acoustic scattering in [3]. The convergence behavior of such kinds of numerical approaches was able to be revealed with the help of the detailed study of the remainder for the error function. Besides, it is noted that numerical analysis with respect to the frequency, which is usually done by solving error equations and extending van der Corput lemma (see [4] p. 333), is able to detect the ability of the numerical method to solve highly oscillatory VIEs. With these techniques in mind, several authors made great contributions to numerical solutions to highly oscillatory VIEs. For example, Galerkin and collocation solutions for VIEs with highly oscillatory trigonometric kernels were investigated in [5,6], highly oscillatory VIEs with weakly singular kernels were studied in [7], the Hermite-type Filon collocation method was presented in [8], and Clenshaw–Curtis–Filon qudrature for Cauchy singular integral equations was investigated in [9].

In this work, we consider the numerical computation of the following second-kind oscillatory VIE:

$$u(t) = f(t) + \int\_0^t K(t, s) \mathbf{e}^{\mathbf{i}\omega \cdot \mathbf{g}(t, s)} u(s) ds, \ t \in [0, T], \tag{1}$$

where *<sup>K</sup>*(*t*,*s*), *<sup>g</sup>*(*t*,*s*) and *<sup>f</sup>*(*t*) are sufficiently smooth, *<sup>u</sup>*(*t*) is unknown, and *<sup>ω</sup>* denotes the oscillation parameter. When *<sup>ω</sup>* = 0, Equation (1) reduces to the classical VIEs. In the case of *<sup>ω</sup>* 1, the kernel in Equation (1) is highly oscillatory, and special quadrature rules should be employed in practical computation.

In the remaining part, we are restricted to the following problems. In the forthcoming section, we first develop a class of generalized multistep collocation methods (*GMCk*1,*k*<sup>2</sup> *M*) for Equation (1) with non-oscillatory kernels, that is, *<sup>ω</sup>* = 0. Then, classical convergence analysis and linear stability analysis are implemented. In the third section, we study the numerical solution to VIE (1) when the kernel changes rapidly, that is, *ω* 1, and present the frequency-explicit convergence analysis. Some concluding remarks are given in Section 4.
