## Abstract and Applied Analysis

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

#### 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 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 $\text{min}\{p,\nu \}$, where $\nu$ 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

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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