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

Haiyan Yuan,Cheng Song

doi:10.1155/2013/679075

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 with0<k<1is proved. For the convergence, the concepts ofD-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 isp, then the method with compound quadrature formula isD-convergent of order at leastmin{p,ν}, whereνdepends on the compound quadrature formula.

Highlights

Volterra delay integro-differential equations (VDIDEs) arise widely in the mathematical modeling of physical and biological phenomena
The class of Runge-Kutta methods with compound quadrature (CQ) formula has been applied to delay-integro-differential equations by many authors (c.f. [18, 19])
It should be pointed out that the adopted quadrature formula (9) is only a class of quadrature formula for Ỹ(in), there exist some other types of quadrature formula, such as Pouzet quadrature (PQ) formula and the quadrature formula based on Laguerre-Radau interpolations [20, 21]

Summary

Introduction

Volterra delay integro-differential equations (VDIDEs) arise widely in the mathematical modeling of physical and biological phenomena. For the case of nonlinear stability and convergence, stability results were obtained in [15, 16], where the authors investigated the nonlinear stability of continuous RungeKutta methods, discrete Runge-Kutta methods, and backward differentiation (BDF) methods, respectively Most of these important results are based on the classical Lipschitz conditions, while the classical Lipschitz conditions are so strong that there are few equations satisfying them. Most of the Volterra delay integro-differential equations satisfy the one-sided Lipschitz condition, while the studies focusing on the stability and convergence of the numerical method for nonlinear VDIDEs based on a one-sided Lipschitz condition have not yet been seen in the literature. These numerical results show that the new methods are quite effective

A Class of VDIDES and the Two-Step Runge-Kutta Methods

The Nonlinear Stability Analysis

The Convergence of TSRK Method for NDDEs

Numerical Experiments

Conclusions