Journal of Applied Mathematics
Volume 2012 (2012), Article ID 456814, 18 pages
http://dx.doi.org/10.1155/2012/456814
Research Article

Some Stability and Convergence of Additive Runge-Kutta Methods for Delay Differential Equations with Many Delays

Haiyan Yuan,1,2 Jingjun Zhao,1 and Yang Xu1

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China

Received 12 October 2011; Accepted 22 December 2011

Academic Editor: Junjie Wei

Copyright © 2012 Haiyan Yuan et al. 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 is devoted to the stability and convergence analysis of the additive Runge-Kutta methods with the Lagrangian interpolation (ARKLMs) for the numerical solution of a delay differential equation with many delays. GDN stability and D-Convergence are introduced and proved. It is shown that strongly algebraically stability gives D-Convergence DA, DAS, and ASI stability give GDN stability. Some examples are given in the end of this paper which confirms our results.