Abstract and Applied Analysis
Volume 2013 (2013), Article ID 679075, 13 pages
http://dx.doi.org/10.1155/2013/679075
Research Article

Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations

Haiyan Yuan1 and Cheng Song2

1Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China
2School of Management, Harbin University of Commerce, Harbin 150028, China

Received 29 January 2013; Revised 11 March 2013; Accepted 15 March 2013

Academic Editor: Changsen Yang

Copyright © 2013 Haiyan Yuan and Cheng Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.