The Floater–Hormann family of the barycentric rational interpolants has recently gained popularity because of its excellent stability properties and highly order of convergence. The purpose of this paper is to design highly accurate and stable schemes based on this family of interpolants for the numerical solution of stiff Volterra integral equations of the second kind.

It is our purpose to introduce a simple multistep method based on linear barycentric rational interpolation for solving ordinary differential equations. Also, we design an adaptive version having one free parameter which is used to improve the stability properties. Numerical experiments of the constructed methods on some well-known stiff problems indicate efficiency and capability of the methods in solving stiff problems.

In this paper, two applications of linear barycentric rational interpolation are used to derive a difference-quadrature scheme for solving a class of systems of Volterra integro-differential equations of the second kind. The convergence of the proposed method is proved, and the order of convergence is obtained in terms of the parameters of the method. Furthermore, the linear stability of the proposed method with respect to both the basic and convolution test equations is analyzed. All the obtained theoretical results are verified by several numerical experiments.