This paper introduces the stability and convergence of two-step Runge-Kutta methods with compound quadrature formula for solving nonlinear Volterra delay integro-differential equations. First, the definitions of -algebraically stable and asymptotically stable are introduced; then the asymptotical stability of a -algebraically stable two-step Runge-Kutta method with is proved. For the convergence, the concepts of -convergence, diagonally stable, and generalized stage order are firstly introduced; then it is proved by some theorems that if a two-step Runge-Kutta method is algebraically stable and diagonally stable and its generalized stage order is , then the method with compound quadrature formula is -convergent of order at least , where depends on the compound quadrature formula.
"Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations." Abstr. Appl. Anal. 2013 (SI01) 1 - 13, 2013. https://doi.org/10.1155/2013/679075