The Stability of Two-Step Runge-Kutta Methods for Neutral Delay Integro Differential-Algebraic Equations with Many Delays

The stability of numerical methods for delay differential equations has been intensively studied in [1–3] for many years.These equations appeared in a wide variety of scientific and engineering fields, such as circuit analysis, computeraided design power systems, and optimal control. The structure for these, the order of convergence, and the asymptotic stability of numerical methods have been studied in [4– 6]. Zhu and Petzold investigated the asymptotic stability of neutral delay differential equations with θ-methods, RungeKutta methods, BDF methods, and linear multistep methods [7]. Zhao et al. studied the stability of neutral delay differential equations with Rosenbrock methods [8]. Yu et al. studied the general neutral delay differential equations withmultistep methods [9]. More recently, there is a growing interest in the analysis of delay integro differential equations. Baker and Ford [10] studied the asymptotic stability of a class of linear multistep (LM) methods for scalar linear delay integro differential equations; Koto [11] dealt with the linear stability of Runge-Kutta (R-K) methods for systems of delay integro differential equations; Huang and Vandewalle [12] gave sufficient andnecessary stability conditions for exact and discrete solutions of linear scalar delay integro differential equations, and Luzyanina et al. [13] developed computational procedures for determining the stability of delay integro differential equations. Zhang and Vandewalle [14] gave the stability criteria for exact anddiscrete solution of neutralmultidelay integro differential equations. Although the stability of numerical methods for delay integro differential equations has been very intensively studied, the stability of delay integro differential equations with many delays has not been studied so far. In this paper, we focus on the asymptotic stability of numerical methods for neutral delay integro differentialalgebraic equations with many delays. This paper is structured as follows. In Section 2 we give asymptotic stability of the analytical solution and introduce two-step Runge-Kutta methods and the stability region. In Section 3, we deal with the asymptotic stability of two-step Runge-Kutta method for neutral delay integro differential-algebraic equations with many delays; the theoretical results are proved. In Section 4, an example is given to illustrate the theoretical results.


Introduction
The stability of numerical methods for delay differential equations has been intensively studied in [1][2][3] for many years.These equations appeared in a wide variety of scientific and engineering fields, such as circuit analysis, computeraided design power systems, and optimal control.The structure for these, the order of convergence, and the asymptotic stability of numerical methods have been studied in [4][5][6].Zhu and Petzold investigated the asymptotic stability of neutral delay differential equations with -methods, Runge-Kutta methods, BDF methods, and linear multistep methods [7].Zhao et al. studied the stability of neutral delay differential equations with Rosenbrock methods [8].Yu et al. studied the general neutral delay differential equations with multistep methods [9].More recently, there is a growing interest in the analysis of delay integro differential equations.Baker and Ford [10] studied the asymptotic stability of a class of linear multistep (LM) methods for scalar linear delay integro differential equations; Koto [11] dealt with the linear stability of Runge-Kutta (R-K) methods for systems of delay integro differential equations; Huang and Vandewalle [12] gave sufficient and necessary stability conditions for exact and discrete solutions of linear scalar delay integro differential equations, and Luzyanina et al. [13] developed computational procedures for determining the stability of delay integro differential equations.Zhang and Vandewalle [14] gave the stability criteria for exact and discrete solution of neutral multidelay integro differential equations.Although the stability of numerical methods for delay integro differential equations has been very intensively studied, the stability of delay integro differential equations with many delays has not been studied so far.
In this paper, we focus on the asymptotic stability of numerical methods for neutral delay integro differentialalgebraic equations with many delays.This paper is structured as follows.In Section 2 we give asymptotic stability of the analytical solution and introduce two-step Runge-Kutta methods and the stability region.In Section 3, we deal with the asymptotic stability of two-step Runge-Kutta method for neutral delay integro differential-algebraic equations with many delays; the theoretical results are proved.In Section 4, an example is given to illustrate the theoretical results.
We know that the stability of analytical solution can be studied via the characteristic equation, so we give a criterion for the asymptotic stability of (1), which is based on the following lemmas.
Where () is a complex function defined by And ln  = ln || +  arg  ( = 0, 1; − < arg  ≤ ) is the principal branch of the multivalued complex natural logarithm.
has at most a finite number of zeros for Re() ≥ .
Since   > 0, we have that Hence, there exist constants Let  be a positive number large enough such that which implies that, for Re() ≥  and || ≥ , That is, p() ̸ = 0 in the set { : Re  ≥ , || ≥ }.By the isolation property of the zeros for analytic functions, p() has at most a finite number of zeros in the set { : Re  ≥ , || < }; this proves the lemma.
Now we will show that the strict inequality in ( 16) holds.Define and then ( Let  be the strictly positive number  = ln(1 + )/; then Thus, the equation P() = 0 has only a finite number of roots when Re() ≥ −, and it holds true for the equation () = 0 by condition (a).Combined with (16) we get that the characteristic equation has at most a finite number of roots in the region { : − ≤ Re() < 0}. Let then  > 0.

The Two-Step Runge-Kutta Methods and the Stability
Region.Consider the two-step Runge-Kutta method: for solving the initial value problem (1).
These methods are a subclass of general linear methods introduced by Butcher [15] and could be possibly also referred to as two-step hybrid methods.They generalize -step collocation methods (with  = 2) for ordinary differential equations (ODEs) studied by Lie and Nørsett [16] and Lie [17] and two-step Runge-Kutta methods for ODEs investigated by Byrne and Lambert [18].The variable stepsize continuous two-step Runge-Kutta methods for ODEs were investigated by Jackiewicz and Tracogna [19].Here we will represent (24a) and (24b) by the following table of the coefficients: where   = ∑  =1 ã and ∑  =1 ( b + b ) = 1 + .Apply (24a) and (24b) to the basic test equation which gives the following equations: Rewriting ( 27) we obtain where To investigate the stability properties of (24a) and ( 24b) with (26), we must investigate the asymptotic behaviors of the solution to (28).This is determined by the location of roots of the characteristic polynomial The stability region of the two-step Runge-Kutta methods (24a) and (24b) is the set of all points  for which the roots of () are inside or on the unit circle with those on the unit circle being simple.If () is a Schur polynomial for any  with Re  < 0, the stability of the two-step Runge-Kutta method contains the negative half plane; the method is said to be A-stable for ODEs.

Asymptotic Stability of TSRK Methods for Neutral Delay Integro Differential-Algebraic Equation with Many Delays
In this section, we will confine our discussion to neutral delay integro differential-algebraic equation with commensurate delays, that is, systems of the form (1) with   = ,  = ℎ,  is a positive integer,  = 1, 2, . . ., .
Definition 6 (see [20]).A numerical method for asymptotically stable system ( 1) is called asymptotically stable if the numerical solution satisfies lim Applying the two-step method (24a) and (24b) to (1), we have We assume that all the eigenvalues of Ã have positive real part.Rearrange the variables of the stage derivatives as Define Rewrite ( 32) and (33) in the form The characteristic polynomial of (37) is given by where Following from the theorem on difference equations, we get that if all the zeros  of (38) satisfy || < 1, then lim Hence, we formulate the following lemmas.
Theorem 9.If the system (( 32) and (33)) satisfies Lemma 8 and the following conditions, then the solution of the TSRK methods for (1) is asymptotically stable.
Proof.By Lemma 7, we need to prove that all the zeros of (38) satisfy || < 1.If these were not true, there would exist a  0 ∈  with By Lemma 8, we have that det[ 1 ( 0 )] ̸ = 0. Hence, (44) is equivalent to Using the Kronecker product [5, chapter 4], we have that Combining ( 45) and (46) gives that This contradicts the assumption that ( b) for | 0 | ≥ 1.Hence, the theorem is proved.Here the matrix coefficients satisfy Theorem 9. Hence, the system is asymptotically stable.

Numerical Experiments
We choose the A-stable TSRK methods as follows [22]: (51) It can be easily seen that the A-stable TSRK method is asymptotically stable, which illuminates the conclusion of Theorem 9.

Conclusions
This paper develops the asymptotic stability of the two-step Runge-Kutta methods for neutral delay integro differentialalgebraic equations with many delays.It studies the asymptotic stability of the analytical solution and introduces two step Runge-Kutta methods and the stability region.It also deals with the asymptotic stability of two-step Runge-Kutta method for neutral delay integro differential-algebraic equations with many delays and proves that the A-stable two-step Runge-Kutta methods are asymptotically stable for neutral delay integro differential-algebraic equations with many delays.