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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

Volterra delay integro-differential equations (VDIDEs) arise widely in the mathematical modeling of physical and biological phenomena. Significant advances in the theoretical analysis and in the numerical analysis for these problems have been made in the last few decades (see, e.g., [

For the case of nonlinear stability and convergence, stability results were obtained in [

The paper is organized as follows. In Section

It is the purpose of this paper to investigate the nonlinear stability and convergence properties of the following initial-value problem VDIDEs:

Throughout this paper, we assume that (

In this paper, we are concerned with two-step Runge-Kutta (TSRK) methods of the form

The class of Runge-Kutta methods with CQ formula has been applied to delay-integro-differential equations by many authors (c.f. [

In this section, we will investigate the stability of the two-step Runge-Kutta methods for VDIDEs. In order to consider the stability property, we also need to consider the perturbed problem of (

Applying the two-step Runge-Kutta method (

For the stability analysis, we need the compound quadrature formula (

Let

Let

We can identify the coefficient matrices

Let

The TSRK method (

Numerical stability is an important feature of an effective numerical method. An unstable numerical method may be consistent of high order, yet arbitrarily small perturbations will eventually cause large deviations from the true solution. In this section, we will focus on the asymptotic stability of the TSRK method.

Assume that the TSRK method (

It follows from a fairly straightforward (but tedious) computation and

The proof of Theorem

Assume that a TSRK method (

Let

The application of Theorem

In order to study the convergence of the method, we define the following:

Method (

TSRK Method (

The concepts of algebraic stability and diagonal stability of TSRK method are the generalizations of corresponding concepts of Runge-Kutta methods. Although it is difficult to examine these conditions, many results have been found, especially, there exist algebraically stable and diagonally stable multistep formulas of arbitrarily high order (cf. [

TSRK Method (

For any given problem (

In this section, we focus on the error analysis of TSRK method for (

First, we give a preliminary result which will later be used several times. To simplify, we denote

Suppose method (

Since the method (

Define

Consider the compact form of (

Suppose the method (

Define

In the following, we assume that the method (

Suppose the method (

A combination of (

Suppose method (

A combination of (

Considering

Consider the following nonlinear Volterra delay integro-differential equations:

Apply the two-step Runge-Kutta method induced by the GL method in [

For Figures

The numerical solution of (

The numerical solution of (

This paper is devoted to the stability and convergence analysis of the two-step Runge-Kutta (TSRK) methods with compound quadrature formula for the numerical solution for a nonlinear Volterra delay integro-differential equations. Nonlinear stability and

We believe that the results presented in this paper can be extended to other general DIDEs and NDIDEs. However, it is difficult to extend the results presented in this paper to more general delay differential equations; the discussion will be discussed later.

This work was supported by the Education Foundation of Heilongjiang Province of China (12523039) and the Doctor Foundation of Heilongjiang Institute of Technology (2012BJ27).