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

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  • Haiyan Yuan
  • Cheng Song

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.

Suggested Citation

  • Haiyan Yuan & Cheng Song, 2013. "Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-13, May.
    • Handle: RePEc:hin:jnlaaa:679075
    DOI: 10.1155/2013/679075
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