Stage-based interpolation Runge-Kutta methods for nonlinear Volterra functional differential equations

In this paper we study stability and convergence of stage-based interpolation Runge-Kutta (SBIRK) methods in which the functional term is approximated by using the stage-values for nonlinear Volterra functional differential equations (VFDEs). This type of methods includes some popular Runge-Kutta methods, for example, some continuous Runge-Kutta methods, as special cases. The uniqueness of solution to discrete algebraic equation is first proved. Unconditional contractivity and asymptotic stability of this class of methods for nonlinear VFDEs are obtained under the most general framework. A global error bound on infinite integration interval for arbitrarily variable step-sizes is also derived for the first time. Some efficient SBIRK methods including Radau IA, Radau IIA and Lobatto IIIC Runge-Kutta methods with appropriate interpolation operator are provided. Numerical results illustrate the effectiveness of the proposed methods for VFDEs and verify our theoretical findings.