Abstract and Applied Analysis

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

Haiyan Yuan and Cheng Song

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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 ( k , l ) -algebraically stable and asymptotically stable are introduced; then the asymptotical stability of a ( k , l ) -algebraically stable two-step Runge-Kutta method with 0 < k < 1 is proved. For the convergence, the concepts of D -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 p , then the method with compound quadrature formula is D -convergent of order at least min { p , ν } , where ν depends on the compound quadrature formula.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 679075, 13 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393444389

Digital Object Identifier
doi:10.1155/2013/679075

Mathematical Reviews number (MathSciNet)
MR3055973

Zentralblatt MATH identifier
1275.65097

Citation

Yuan, Haiyan; Song, Cheng. Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 679075, 13 pages. doi:10.1155/2013/679075. https://projecteuclid.org/euclid.aaa/1393444389


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