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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 456814, 18 pages
Research Article

Some Stability and Convergence of Additive Runge-Kutta Methods for Delay Differential Equations with Many Delays

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China

Received 12 October 2011; Accepted 22 December 2011

Academic Editor: Junjie Wei

Copyright © 2012 Haiyan Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper is devoted to the stability and convergence analysis of the additive Runge-Kutta methods with the Lagrangian interpolation (ARKLMs) for the numerical solution of a delay differential equation with many delays. GDN stability and D-Convergence are introduced and proved. It is shown that strongly algebraically stability gives D-Convergence DA, DAS, and ASI stability give GDN stability. Some examples are given in the end of this paper which confirms our results.

1. Introduction

Delay differential equations arise in a variety of fields as biology, economy, control theory, electrodynamics (see, e.g., [15]). When considering the applicability of numerical methods for the solution of DDEs, it is necessary to analyze the stability of the numerical methods. In the last three decades, many works had dealt with these problems (see, e.g., [6]). For the case of nonlinear delay differential equations, this kind of methodology had been first introduced by Torelli [7, 8] and then developed by Bellen and Zennaro [9], Bellen [10], and Zennaro [11, 12].

In this paper, we consider the following nonlinear DDEs with 𝑚 delays: 𝑦(𝑡)=𝑓[1]𝑡,𝑦(𝑡),𝑦𝑡𝜏1+𝑓[2]𝑡,𝑦(𝑡),𝑦𝑡𝜏2++𝑓[𝑚]𝑡,𝑦(𝑡),𝑦𝑡𝜏𝑚𝑡𝑡0,𝑡,𝑇𝑦(𝑡)=𝜑(𝑡)𝑡0𝜏,𝑡0,(1.1)𝑧(𝑡)=𝑓[1]𝑡,𝑧(𝑡),𝑧𝑡𝜏1+𝑓[2]𝑡,𝑧(𝑡),𝑧𝑡𝜏2++𝑓[𝑚]𝑡,𝑧(𝑡),𝑧𝑡𝜏𝑚𝑡𝑡0𝑡,𝑇𝑧(𝑡)=𝜓(𝑡)𝑡0𝜏,𝑡0,(1.2) where 𝜏1𝜏2𝜏𝑚=𝜏, 𝑓[𝑣][𝑡0,𝑇]×𝐶𝑁×𝐶𝑁𝐶𝑁, 𝑣=1,2,,𝑚, and 𝜑,𝜓[𝑡0𝜏,𝑡0]𝐶𝑁 are continuous functions such that (1.1) and (1.2) have a unique solution, respectively. Moreover, we assume that there exist some inner product , and the induced norm such that𝑓Re[𝑣]𝑡,𝑦1,𝑢𝑓[𝑣]𝑡,𝑦2,𝑢,𝑦1𝑦2𝜎𝑣𝑦1𝑦22𝑓𝑣=1,2,,𝑚,[𝑣]𝑡,𝑦,𝑢1𝑓[𝑣]𝑡,𝑦,𝑢2𝑟𝑣𝑢1𝑢2𝑣=1,2,,𝑚,(1.3)forall𝑡[𝑡0,𝑇],forall𝑦,𝑦1,𝑦2,𝑢,𝑢1,𝑢2𝐶𝑁, where 𝜎𝑣, 𝑟𝑣 are constants with0𝑟𝑣𝜎𝑣,𝑣=1,2,,𝑚.(1.4) Space discretization of some time dependent delay partial differential equations give rise to such delay differential equations containing additive terms with different stiffness properties. In these situations, additive Runge-Kutta (ARK) methods are used. Some recent works about ARK can refer to [13, 14]. For the additive DDEs (1.1), (1.2), similar to the proof of Theorem 2.1 in [7], it is straightforward to prove that under the conditions (1.3) and (1.4), the analytic solutions satisfy𝑦(𝑡)𝑧(𝑡)max𝑡0𝜏𝑡𝑡0𝜑(𝑡)𝜓(𝑡).(1.5) To demand the discrete numerical solutions to preserve the stability properties (1.5) of the analytic solutions, Torelli [7] introduced a concept of RN, GRN stability for numerical methods applied to dissipative nonlinear systems of DDEs such as (1.1), which is the straightforward generalization of the well-known concept of BN stability of numerical methods with respect to dissipative systems of ODEs (see also [9]). A disappointing conclusion is, as it is described in [10], that the order of RK methods for DDEs preserving RN-stable properties may not be more than 4.

To bypass this order barrier, Zhang and Zhou [15] relaxed the RN stability restriction, considered the GDN stability and D-Convergence of (1.1) in the case 𝑚=1. In 2001, Zhang et al. [16] gave the results of D-Convergence and GDN stability of (1.1) with the vector form. So, the aim of this paper is the study of stability and convergence properties for ARK methods when they are applied to nonlinear delay differential equations with 𝑚 delays.

2. The GDN Stability of the Additive Runge-Kutta Method

In this preparatory section we recall the additive Runge-Kutta method and give out its stability analysis.

Definition 2.1. An additive Runge-Kutta method with the Lagrangian interpolation (ARKLM) of 𝑠 stages and 𝑚 levels for (1.1) is a one-step numerical method which the numerical solution of (1.1) from 𝑦𝑛 (numerical approximation at 𝑡𝑛) to 𝑦𝑛+1 (numerical approximation at 𝑡𝑛+1=𝑡𝑛+), that is, 𝑦𝑛+1=𝑦𝑛+𝑚𝑠𝑣=1𝑗=1𝑏𝑗[𝑣]𝑓[𝑣]𝑡𝑛+𝑐𝑗,𝑦𝑗(𝑛),̃𝑦𝑗[𝑣](𝑛)𝑦𝑖(𝑛)=𝑦𝑛+𝑚𝑠𝑣=1𝑗=1𝑎[𝑣]𝑖𝑗𝑓[𝑣]𝑡𝑛+𝑐𝑗,𝑦𝑗(𝑛),̃𝑦𝑗[𝑣](𝑛)𝑖=1,2,,𝑠,𝑣=1,2,,𝑚.(2.1a) Here the coefficients 𝑎[𝑣]𝑖𝑗, 𝑏𝑗[𝑣], and 𝑐𝑗 satisfy 𝑠𝑗=1𝑎[𝑣]𝑖𝑗=𝑐𝑗[𝑣],0𝑐𝑗1,𝑗,𝑣=1,2,,𝑚,(2.1b)𝑡𝑛=𝑡0+𝑛, 𝑦𝑛, 𝑦𝑗(𝑛), ̃𝑦𝑗[𝑣](𝑛) are approximations to the analytic solution 𝑦(𝑡𝑛), 𝑦(𝑡𝑛+𝑐𝑗), 𝑦(𝑡𝑛+𝑐𝑗𝜏𝑣) of (1.1), respectively, and the argument ̃𝑦𝑗[𝑣](𝑛) is determined by ̃𝑦𝑗[𝑣](𝑛)=𝜑𝑡𝑛+𝑐𝑗𝜏𝑣𝑡𝑛+𝑐𝑗𝜏𝑣0,𝑟𝑃𝑣=𝑑𝐿𝑃𝑣𝛿𝑣𝑦(𝑛𝑚𝑣+𝑃𝑣)𝑗𝑡𝑛+𝑐𝑗𝜏𝑣>0.(2.1c) With 𝜏𝑣=(𝑚𝑣𝛿𝑣), 𝛿𝑣[0,1), integer 𝑚𝑣𝑟+1, 𝑟,𝑑0, and 𝐿𝑃𝑣𝛿𝑣=𝑟𝑘=𝑑𝑘𝑃𝑣𝛿𝑣𝑘𝑃𝑣𝑃𝑘𝑣=𝑑,𝑑+1,,𝑟.(2.2) We assume 𝑚𝑣𝑟+1 is to guarantee that no (unknown) values 𝑦𝑗(𝑖) with 𝑖𝑛 are used in the interpolation procedure. In addition, we always put 𝑦𝑗(𝑖)=𝜑(𝑡𝑛+𝑐𝑗) whenever 𝑛<0, and 𝑦𝑛=𝜑(𝑡𝑛) whenever 𝑛0.

The coefficients of the method may be organized in the Butcher tableau𝐶𝐴[1]𝐴[2]𝐴[𝑚]𝑏[1]𝑇𝑏[2]𝑇𝑏[𝑚]𝑇,(2.3) where 𝐶=[𝑐1,𝑐2,,𝑐𝑠]𝑇 and for 𝑣=1,2,,𝑚,𝑏[𝑣]=𝑏1[𝑣],𝑏2[𝑣],,𝑏𝑠[𝑣],𝐴[𝑣]=𝑎[𝑣]𝑖𝑗𝑠𝑖,𝑗=1.(2.4) In order to write (2.1a), (2.1b), and (2.1c) in a more compact way we introduce some notations. The 𝑁×𝑁 identity matrix will be denoted by 𝐼𝑁, 𝑒=(1,1,,1)𝑇𝑅𝑆, 𝐺=𝐺𝐼𝑁 is the Kronecker product of matrix 𝐺 and 𝐼𝑁. For 𝑢=(𝑢1,𝑢2,,𝑢𝑠)𝑇,𝑣=(𝑣1,𝑣2,,𝑣𝑠)𝑇𝐶𝑁𝑆, we define the inner product and the induced norm in 𝐶𝑁𝑆 as follows:𝑢,𝑣=𝑠𝑖=1𝑢𝑖,𝑣𝑖,𝑢=𝑠𝑖=1𝑢𝑖2.(2.5)

Moreover, we also adopt that𝑦(𝑛)=𝑦1(𝑛)𝑦2(𝑛)𝑦𝑠(𝑛),̃𝑦[𝑣](𝑛)=̃𝑦1[𝑣](𝑛)̃𝑦2[𝑣](𝑛)̃𝑦𝑠[𝑣](𝑛),𝑓[𝑣]𝑡𝑛,𝑦(𝑛),̃𝑦[𝑣](𝑛)=𝑓[𝑣]𝑡𝑛,𝑦1(𝑛),̃𝑦1[𝑣](𝑛)𝑓[𝑣]𝑡𝑛,𝑦2(𝑛),̃𝑦2[𝑣](𝑛)𝑓[𝑣]𝑡𝑛,𝑦𝑠(𝑛),̃𝑦𝑠[𝑣](𝑛).(2.6) With the above notation, method (2.1a), (2.1b), and (2.1c) can be written as 𝑦𝑛+1=𝑦𝑛+𝑚𝑣=1̃𝑏[𝑣]𝑇𝑓[𝑣]𝑡𝑛,𝑦(𝑛),̃𝑦[𝑣](𝑛),𝑦(𝑛)=̃𝑒𝑦𝑛+𝑚𝑣=1𝐴[𝑣]𝑓[𝑣]𝑡𝑛,𝑦(𝑛),̃𝑦[𝑣](𝑛),̃𝑦[𝑣](𝑛)=𝑡̃𝑒𝜑𝑛+𝑐𝑗𝜏𝑣,𝑡𝑛+𝑐𝑗𝜏𝑣𝑡0,𝑟𝑃𝑣=𝑑𝐿𝑃𝑣𝛿𝑣𝑦(𝑛𝑚𝑣+𝑃𝑣),𝑡𝑛+𝑐𝑗𝜏𝑣>𝑡0.(2.7) In 1997, Zhang and Zhou [15] introduced the extension of RN stability to GDN stability as follows.

Definition 2.2. An ARKLM (2.1a), (2.1b), and (2.1c) for DDEs is called GDN stable if, under the conditions (1.3) and (1.4), numerical approximations 𝑦𝑛 and 𝑧𝑛 to the solution of (1.1) and (1.2), respectively, satisfy 𝑦𝑛𝑧𝑛𝐶max𝑡0𝜏𝑡𝑡0𝜑(𝑡)𝜓(𝑡),𝑛0,(2.8) where constant 𝐶>0 depends only on the method, the parameter 𝜎𝑣, 𝑣=1,2,,𝑚, and the interval length 𝑇𝑡0.

Here, we can see the constant 𝐶 need not to be less than 1, otherwise the Definition 2.2 is just RN stable in [7].

Definition 2.3. An ARKLM (2.1a), (2.1b), and (2.1c) is called strongly algebraically stable if matrices 𝑀𝛾𝜇 are nonnegative definite, where 𝑀𝛾𝜇=𝐵[𝛾]𝐴[𝜇]+𝐴𝑇[𝛾]𝐵[𝜇]𝑏[𝛾]𝑏𝑇[𝜇],𝐵[𝛾]𝑏=diag1[𝛾],𝑏2[𝛾],,𝑏𝑠[𝛾],(2.9) for 𝜇,𝛾=1,2,,𝑚.

Let {𝑦𝑛,𝑦𝑗(𝑛),̃𝑦𝑗[1](𝑛),̃𝑦𝑗[2](𝑛),,̃𝑦𝑗[𝑚(𝑛)]}𝑠𝑗=1 and {𝑧𝑛,𝑧𝑗(𝑛),̃𝑧𝑗[1](𝑛),̃𝑧𝑗[2](𝑛),,̃𝑧𝑗[𝑚](𝑛)}𝑠𝑗=1 be two sequences of approximations to problems (1.1) and (1.2), respectively. From method (2.1a), (2.1b), and (2.1c) with the samestep size , and write𝑇𝑖(𝑛)=𝑡𝑛+𝑐𝑖,𝑈𝑖(𝑛)=𝑦𝑖(𝑛)𝑧𝑖(𝑛),𝑈𝑖[𝑣](𝑛)=̃𝑦𝑖[𝑣](𝑛)̃𝑧𝑖[𝑣](𝑛),𝑈0(𝑛)=𝑦𝑛𝑧𝑛,𝑄𝑖[𝑣](𝑛)𝑓=[𝑣]𝑇𝑖(𝑛),𝑦𝑖(𝑛),̃𝑦𝑖[𝑣](𝑛)𝑓[𝑣]𝑇𝑖(𝑛),𝑧𝑖(𝑛),̃𝑧𝑖[𝑣](𝑛),𝑖=1,2,,𝑠,𝑣=1,2,,𝑚.(2.10) Then (2.1a) reads 𝑈0(𝑛+1)=𝑈0(𝑛)+𝑚𝑠𝑣=1𝑖=1𝑏𝑖[𝑣]𝑄𝑖[𝑣](𝑛),𝑈𝑖(𝑛)=𝑈0(𝑛)+𝑚𝑠𝑣=1𝑗=1𝑎[𝑣]𝑖𝑗𝑄𝑗[𝑣](𝑛).(2.11) Our main results about GDN stability are contained in the following theorem.

Theorem 2.4. Assume ARK method (2.1a) is strongly algebraically stable, and then the corresponding ARKLM (2.1a), (2.1b), and (2.1c) is GDN stable, and satisfies 𝑦𝑛𝑧𝑛exp𝑇𝑡0𝑚𝑚𝑣=1𝜎𝑣𝐿0max𝑡0𝜏𝑡𝑡0𝜑(𝑡)𝜓(𝑡)2,𝑛0,(2.12) where 𝐿0=sup𝛿𝜈[0,1)(𝑟𝑝𝜈=𝑑|𝐿𝑝𝜈(𝛿𝜈)|)2.

Proof. From (2.11) we get 𝑈0(𝑛+1)2=𝑈0(𝑛)+𝑚𝑠𝑣=1𝑖=1𝑏𝑖[𝑣]𝑄𝑖[𝑣](𝑛),𝑈0(𝑛)+𝑚𝑠𝑣=1𝑖=1𝑏𝑖[𝑣]𝑄𝑖[𝑣](𝑛)=𝑈0(𝑛)2+2𝑚𝑠𝑣=1𝑖=1𝑏𝑖[𝑣]𝑄Re𝑖[𝑣](𝑛),𝑈0(𝑛)+𝑚𝑠𝑢,𝑣=1𝑖,𝑗=1𝑏𝑖[𝑢]𝑏𝑗[𝑣]𝑄𝑖[𝑢](𝑛),𝑄𝑗[𝑣](𝑛)=𝑈0(𝑛)2+2𝑚𝑠𝑣=1𝑖=1𝑏𝑖[𝑣]𝑄Re𝑖[𝑣](𝑛),𝑈𝑖(𝑛)𝑚𝑠𝑣=1𝑗=1𝑎[𝑣]𝑖𝑗𝑄𝑗[𝑣](𝑛)+𝑚𝑠𝑢,𝑣=1𝑖,𝑗=1𝑏𝑖[𝑢]𝑏𝑗[𝑣]𝑄𝑖[𝑣](𝑛),𝑄𝑗[𝑢](𝑛)=𝑈0(𝑛)2+2𝑚𝑠𝑣=1𝑖=1𝑏𝑖[𝑣]𝑄Re𝑖[𝑣](𝑛),𝑈𝑖(𝑛)𝑚𝑠𝑢,𝑣=1𝑖,𝑗=1𝑏𝑖[𝑢]𝑎[𝑣]𝑖𝑗+𝑏𝑗[𝑣]𝑎[𝑢]𝑖𝑗𝑏𝑖[𝑢]𝑏𝑗[𝑣]𝑄𝑖[𝑣](𝑛),𝑄𝑗[𝑢](𝑛).(2.13) If the matrices 𝑀𝛾𝜇 are nonnegative definite, then 𝑈0(𝑛+1)2𝑈0(𝑛)2+2𝑚𝑠𝑣=1𝑖=1𝑏𝑖[𝑣]𝑄Re𝑖[𝑣](𝑛),𝑈𝑖(𝑛).(2.14) Furthermore, by conditions (1.3) and (1.4) and Schwartz inequality we have 𝑄Re𝑗[𝑣](𝑛),𝑈𝑗(𝑛)𝑓=[𝑣]𝑇𝑗(𝑛),𝑦𝑗(𝑛),̃𝑦𝑗[𝑣](𝑛)𝑓[𝑣]𝑇𝑗(𝑛),𝑧𝑗(𝑛),̃𝑧𝑗[𝑣](𝑛),𝑈𝑗(𝑛)𝑓=Re[𝑣]𝑇𝑗(𝑛),𝑦𝑗(𝑛),̃𝑦𝑗[𝑣](𝑛)𝑓[𝑣]𝑇𝑗(𝑛),𝑧𝑗(𝑛),̃𝑦[𝑣](𝑛),𝑈𝑗(𝑛)𝑓+Re[𝑣]𝑇𝑗(𝑛)𝑧𝑗(𝑛),̃𝑦[𝑣](𝑛)𝑓[𝑣]𝑇𝑗(𝑛),𝑧𝑗(𝑛),̃𝑧𝑗(𝑛),𝑈𝑗(𝑛)𝜎𝑣𝑈𝑗(𝑛)2𝑓+[𝑣]𝑇𝑗(𝑛),𝑧𝑗(𝑛),̃𝑦𝑗[𝑣](𝑛)𝑓[𝑣]𝑇𝑗(𝑛),𝑧𝑗(𝑛),̃𝑧𝑗[𝑣](𝑛)𝑈𝑗(𝑛)𝜎𝑣𝑈𝑗(𝑛)2+𝑟𝑣𝑈𝑗[𝑣](𝑛)𝑈𝑗(𝑛)𝜎𝑣𝑈𝑗(𝑛)2+12𝑟𝑣𝑈𝑗[𝑣](𝑛)2+𝑈𝑗(𝑛)2.(2.15) From (1.4), we know 0𝑟𝑣𝜎𝑣.
Then, we have 𝑄Re𝑗[𝑣](𝑛),𝑈𝑗(𝑛)𝜎𝑣𝑈𝑗(𝑛)212𝑟𝑣𝑈𝑗[𝑣](𝑛)2+𝑈𝑗(𝑛)212𝜎𝑣𝑈𝑗[𝑣](𝑛)2.(2.16) Substituting (2.16) into (2.14), yields 𝑈0(𝑛+1)2𝑈0(𝑛)2𝑚𝑠𝑣=1𝑗=1𝜎𝑣𝑏𝑗[𝑣]𝑈𝑗[𝑣](𝑛)2.(2.17) In addition, with (2.1c), we have 𝑈𝑗[𝑣](𝑛)2𝑟𝑃𝑣=𝑑||𝐿𝑃𝑣𝛿𝑣||𝑈(𝑛𝑚𝑣+𝑃𝑣)𝑗2𝐿0𝑈max(𝑛𝑚𝑣+𝑃𝑣)𝑗2.(2.18) Combining (2.17) with (2.18) and using (2.1b) we arrive at 𝑈0(𝑛+1)21𝑚𝑠𝑣=1𝑗=1𝑏𝑗[𝑣]𝜎𝑣𝐿0𝑈max0(𝑛)2,max(𝑗,𝑃𝑣)𝐸𝑈(𝑛𝑚𝑣+𝑃𝑣)𝑗21𝑚𝑚𝑣=1𝜎𝑣𝐿0𝑈max0(𝑛)2,max(𝑗,𝑃𝑣)𝐸𝑈(𝑛𝑚𝑣+𝑃𝑣)𝑗2,(2.19) where 𝐸={(𝑗,𝑃𝑣)1𝑗𝑠,𝑑𝑃𝑣𝑟}.
Similar to (2.19), the inequalities 𝑈𝑖(𝑛)1𝑚𝑚𝑣=1𝜎𝑣𝐿0𝑈max0(𝑛)2,max𝑗,𝑃𝑣𝐸𝑈(𝑛𝑚𝑣+𝑃𝑣)𝑗2𝑖=1,2,,𝑠,(2.20) follow.
In the following, with the help of inequalities (2.19), (2.20) and induction we will prove the inequalities: 𝑈𝑖(𝑛)21𝑚𝑚𝑣=1𝜎𝑣𝐿0𝑛+1max𝑡0𝜑(𝑡)𝜓(𝑡)2,𝑛0,𝑖=1,2,,𝑠.(2.21) In fact, it is clear from (2.19), (2.20), and 𝑚𝑣𝑟+1 that 𝑈𝑖(0)21𝑚𝑚𝑣=1𝜎𝑣𝐿0max𝑡0𝜑(𝑡)𝜓(𝑡)2𝑖=0,1,2,,𝑠.(2.22) Suppose for 𝑛𝑘(𝑘0) that 𝑈𝑖(𝑛)21𝑚𝑚𝑣=1𝜎𝑣𝐿0𝑛+1max𝑡0𝜑(𝑡)𝜓(𝑡)2,𝑖=0,1,2,,𝑠.(2.23) Then from (2.19), (2.20), 𝑚𝑣𝑟+1, and (1𝑚𝑣=1𝜎𝑣𝐿0)>1, we conclude that 𝑈𝑖(𝑘+1)21𝑚𝑚𝑣=1𝜎𝑣𝐿0𝑘+2max𝑡0𝜏𝑡𝑡0𝜑(𝑡)𝜓(𝑡)2,𝑖=0,1,2,,𝑠.(2.24) This completes the proof of inequalities (2.21). In view of (2.21), we get for 𝑛0 that 𝑈0(𝑛)21𝑚𝑚𝑣=1𝜎𝑣𝐿0𝑛+1max𝑡0𝜏𝑡𝑡0𝜑(𝑡)𝜓(𝑡)2exp(𝑛+1)𝑚𝑚𝑣=1𝜎𝑣𝐿0max𝑡0𝜏𝑡𝑡0𝜑(𝑡)𝜓(𝑡)2exp𝑇𝑡0𝑚𝑚𝑣=1𝜎𝑣𝐿0max𝑡0𝜏𝑡𝑡0𝜑(𝑡)𝜓(𝑡)2.(2.25) As a result, we know that method (2.1a), (2.1b), and (2.1c) is GDN stable.

3. D-Convergence

In order to study the convergence of numerical methods for DDEs, we have to mention the concept of the convergence for stiff ODEs.

In 1981, Frank et al. [17] introduced the important concept of B-convergence for numerical methods applied to nonlinear stiff initial value problems of ordinary differential equations. Later, there have been rapid developments in the study of B-convergence and a significant number of important results have already been found for Runge-Kutta methods. In fact, B-convergence result is nothing but a realistic global error estimate based on one-sided Lipschitz constant [18]. In this section, we start discussing the convergence of ARKLM (2.1a), (2.1b), and (2.1c) for DDEs (1.1) with conditions (1.3) and (1.4). The approach to the derivation of these estimates is similar to that used in [15]. We assume the analytic solution 𝑦(𝑡) of (1.1) is smooth enough and its derivatives used later are bounded by𝐷(𝑖)𝑀𝑦(𝑡)𝑖𝑡𝑡0𝜏,𝑇,(3.1) where 𝐷(𝑖)𝑦𝑦(𝑡)=(𝑖)𝑡(𝑡),𝑡0+(𝑗1),𝑡0,𝑦+𝑗(𝑖)𝑡0+𝑗0,𝑡=𝑡0+𝑗.(3.2) If we introduce some notations𝑌(𝑛)=𝑦𝑡𝑛+𝑐1𝑦𝑡𝑛+𝑐2𝑦𝑡𝑛+𝑐𝑠,𝑌[𝑣](𝑛)=𝑦𝑡𝑛+𝑐1𝜏𝑣𝑦𝑡𝑛+𝑐2𝜏𝑣𝑦𝑡𝑛+𝑐𝑠𝜏𝑣,(3.3) with the above notations, the local errors in (2.7) can be defined as𝑦𝑡𝑛+1𝑡=𝑦𝑛+𝑚𝑣=1̃𝑏[𝑣]𝑇𝑓[𝑣]𝑡𝑛,𝑌(𝑛),𝑌[𝑣](𝑛)+𝑄𝑛,(3.4a)𝑌(𝑛)𝑡=̃𝑒𝑦𝑛+𝑚𝑣=1𝐴[𝑣]𝑓[𝑣]𝑡𝑛,𝑌(𝑛),𝑌[𝑣](𝑛)+𝑟𝑛,(3.4b)𝑌[𝑣](𝑛)𝑌=(1[𝑣](𝑛),𝑌2[𝑣](𝑛)𝑌,,𝑠[𝑣](𝑛))𝑇 with 𝑌𝑗[𝑣](𝑛)=𝜑𝑡𝑛+𝑐𝑗𝜏𝑣𝑡𝑛+𝑐𝑗𝜏𝑣𝑡0,𝑟𝑃𝑣=𝑑𝐿𝑃𝑣𝛿𝑣𝑦(𝑛𝑚𝑣+𝑃𝑣)𝑗+𝜌𝑗[𝑣](𝑛)𝑡𝑛+𝑐𝑗𝜏𝑣>𝑡0.(3.4c)If we take ̆𝑦𝑛=𝑦(𝑡𝑛) and̆𝑦(𝑛)=𝑦𝑡𝑛+𝑐1𝑦𝑡𝑛+𝑐2𝑦𝑡𝑛+𝑐𝑠̆𝑦[𝑣](𝑛)=𝑦𝑡𝑛+𝑐1𝜏𝑣𝑦𝑡𝑛+𝑐2𝜏𝑣𝑦𝑡𝑛+𝑐𝑠𝜏𝑣.(3.5) Then we can get the perturbed scheme of (2.7) ̃𝑦𝑛+1=̃𝑦𝑛+𝑚𝑣=1̃𝑏[𝑣]𝑇𝑓[𝑣]𝑡𝑛,̃𝑦(𝑛),̆̃𝑦[𝑣](𝑛)+𝑄𝑛,(3.6a)̃𝑦(𝑛)=̃𝑒̃𝑦𝑛+𝑚𝑣=1𝐴[𝑣]𝑓[𝑣]𝑡𝑛,̃𝑦(𝑛),̆̃𝑦[𝑣](𝑛)+𝑟𝑛,(3.6b)̆̃𝑦[𝑣](𝑛)=𝑡̃𝑒𝜑𝑛+𝑐𝑗𝜏𝑣,𝑡𝑛+𝑐𝑗𝜏𝑣0,𝑟𝑃𝑣=𝑑𝐿𝑃𝑣𝛿𝑣̆𝑦(𝑛𝑚𝑣+𝑃𝑣)+𝜌[𝑣](𝑛),𝑡𝑛+𝑐𝑗𝜏𝑣>0.(3.6c)With perturbations𝑄𝑛𝐶𝑁,𝑟𝑛=𝑟1𝑇(𝑛),𝑟2𝑇(𝑛),,𝑟𝑠𝑇(𝑛)𝑇,𝜌[𝑣](𝑛)=𝜌1𝑇[𝑣](𝑛),𝜌2𝑇[𝑣](𝑛),,𝜌𝑠𝑇[𝑣](𝑛)𝐶𝑁𝑆.(3.7) According to Taylor formula and the formula in [19, pages 205–212], 𝑄𝑛,𝑟𝑛, and 𝜌[𝑣](𝑛) can be determined, respectively, as following:𝑄𝑛=𝑃𝑙=1𝑙1(𝑙1)!𝑙𝑚𝑠𝑣=1𝑗=1𝑏𝑗[𝑣]𝑐𝑗𝑙1𝐷(𝑙)𝑦𝑡𝑛+𝑅0(𝑛),𝑟𝑖(𝑛)=𝑃𝑙=1𝑙1(𝑙1)!𝑙𝑐𝑙𝑖𝑚𝑠𝑣=1𝑗=1𝑎[𝑣]𝑖𝑗𝑐𝑗𝑙1𝐷(𝑙)𝑦𝑡𝑛+𝑅𝑖(𝑛),𝜌𝑖[𝑣](𝑛)=𝑞+1(𝑞+1)!𝑟𝑃𝑣=𝑑𝛿𝑣𝑃𝑣𝐷(𝑞+1)𝑦𝜉𝑖(𝑛),𝜉𝑖(𝑛)𝑡𝑛𝑚𝑣𝑑+𝑐𝑖,𝑡𝑛𝑚𝑣+𝑟+𝑐𝑖,(3.8) where 𝑞=𝑑+𝑟, 𝑅𝑖(𝑛), and 𝜉𝑖(𝑛) satisfy 𝑅𝑖(𝑛)𝑀𝑖𝑖+1, 𝑖=0,1,2,,𝑠, (0,0], 0 depends only on the method, and 𝑀𝑖(𝑖=0,1,2,,𝑠) depends only on the method and some 𝑀𝑖 in (3.2).

Combining (2.7) with (3.6a), (3.6b), and (3.6c) yields the following recursion scheme for the 𝜀0(𝑛+1)=̆𝑦𝑛+1𝑦𝑛+1:𝜀0(𝑛+1)=𝜀0(𝑛)+𝑚𝑣=1̃𝑏[𝑣]𝑇𝑓[𝑣]𝑡𝑛,𝑦𝑛,̆̃𝑦[𝑣](𝑛)𝑓[𝑣]𝑡,𝑦𝑛,̃𝑦[𝑣](𝑛)+𝑔[𝑣](𝑛)𝜀𝑛+𝑄𝑛,𝜀𝑛=̃𝑒𝜀0(𝑛)+𝑚𝑣=1𝐴[𝑣]𝑓[𝑣]𝑡𝑛,𝑦𝑛,̆̃𝑦[𝑣](𝑛)𝑓[𝑣]𝑡,𝑦𝑛,̃𝑦[𝑣](𝑛)+𝑔[𝑣](𝑛)𝜀𝑛+𝑟𝑛,(3.9) where 𝜀0(𝑛+1)=̆𝑦𝑛+1𝑦𝑛+1, 𝜀𝑛=(𝜀1𝑇(𝑛),𝜀2𝑇(𝑛),,𝜀𝑠𝑇(𝑛))𝑇=̆𝑦(𝑛)𝑦(𝑛) and 𝑔𝑖[𝑣](𝑛)=10𝑓2(𝑡𝑛+𝑐𝑖,𝑦𝑖(𝑛)+𝜃(̃𝑦𝑖(𝑛)𝑦𝑖(𝑛)̆),̃𝑦[𝑣](𝑛))𝑑𝜃, 𝑖=1,2,,𝑠, 𝑓2(𝑡,𝑢,𝑣) is the Jacobian matrix (𝜕𝑓(𝑡,𝑢,𝑣)/𝜕𝑢)(𝑡𝑅,𝑢,𝑣𝐶𝑁).

Assume that (𝐼𝑠𝑚𝑣=1𝐴[𝑣]𝑔[𝑣](𝑛)) is regular, from (3.9), we can get𝜀0(𝑛+1)=𝐼𝑁+𝑚𝑣=1̃𝑏[𝑣]𝑇𝐼𝑠𝑚𝑣=1𝐴[𝑣]𝑔[𝑣](𝑛)1̃𝑒𝑔[𝑣](𝑛)𝜀0(𝑛)+𝑚𝑣=1̃𝑏[𝑣]𝑇𝑔[𝑣](𝑛)𝐼𝑠𝑚𝑣=1𝐴[𝑣]𝑔[𝑣](𝑛)1𝑟𝑛+𝑚𝑣=1̃𝑏[𝑣]𝑇𝐼𝑠+𝐼𝑠𝑚𝑣=1𝐴[𝑣]𝑔[𝑣](𝑛)1𝑚𝑣=1𝐴[𝑣]𝑔[𝑣](𝑛)𝑓[𝑣]𝑡𝑛,𝑦(𝑛),̆̃𝑦[𝑣](𝑛)𝑓[𝑣]𝑡,𝑦(𝑛),̃𝑦[𝑣](𝑛)+𝑄𝑛.(3.10) Now, we introduce the concept of D-Convergence from [15].

Definition 3.1. An ARKLM (2.1a), (2.1b), and (2.1c) with 𝑦𝑛=𝑦(𝑡𝑛)(𝑛0), 𝑦𝑖(𝑛)=𝑦(𝑡𝑛+𝑐𝑖)(𝑛<0) and ̃𝑦𝑖[𝑣](𝑛)=𝑦(𝑡𝑛+𝑐𝑖𝜏𝑣)(𝑛<0) is called D-Convergence of order 𝑝 if this method, when applied to any given DDEs (1.1) subject to (1.3) and (1.4), produces an approximation sequence 𝑦𝑛, and the global error satisfies a bound of the form 𝑦𝑡𝑛𝑦𝑛𝑡𝐶𝑛𝑃,0,0,(3.11) where the maximum stepsize 0 depends on characteristic parameter 𝜎𝑣 and the method, the function 𝐶(𝑡) depends only on some 𝑀𝑖 in (3.2), delay 𝜏𝑣, characteristic parameters 𝜎𝑣, 𝑟𝑣 and the method, 𝑣=1,2,,𝑚.

Definition 3.2. The ARKLM (2.1a), (2.1b), and (2.1c) is said to be DA stable if the matrix (𝐼𝑠𝑚𝜈=1𝐴[𝜈]𝜉) is regular for 𝜉𝐶={𝜉𝐶Re𝜉0} and |𝑅𝑖(𝜉)|1, for all 𝜉𝐶, 𝑖=0,1,,𝑠, where 𝑅𝑖𝜀1=1+𝑚𝑣=1𝐴𝑖[𝑣]𝜀1𝐼𝑠𝑚𝜈=1𝐴[𝑣]𝜉1𝐴𝑒,0[𝑣]=𝑏[𝑣],𝐴𝑖[𝑣]=𝑎[𝑣]𝑖1,𝑎[𝑣]𝑖2,,𝑎[𝑣]𝑖𝑠𝑇,𝑖=0,1,,𝑠.(3.12)

Definition 3.3. The ARKLM (2.1a), (2.1b), and (2.1c) is said to be ASI stable if the matrix (𝐼𝑠𝑚𝜈=1𝐴[𝜈]𝜉) is regular for 𝜉𝐶, and (𝐼𝑠𝑚𝜈=1𝐴[𝜈]𝜉)1 is uniformly bounded for 𝜉𝐶.

Definition 3.4. The ARKLM (2.1a), (2.1b), and (2.1c) is said to be DAS stable if the matrix (𝐼𝑠𝑚𝜈=1𝐴[𝜈]𝜉) is regular for 𝜉𝐶, and 𝑚𝜈=1𝐴[𝑣]𝑇𝑖𝜉(𝐼𝑠𝑚𝜈=1𝐴[𝜈]𝜉)1(𝑖=0,1,,𝑠) is uniformly bounded for 𝜉𝐶.

Lemma 3.5. Suppose the ARKLM is DA, DAS, and ASI stable, then there exist positive constants 0, 𝛾1, 𝛾2, 𝛾3, which depend only on the method and the parameter 𝜎𝜈, 𝑟𝜈, such that 𝐼𝑠𝑚𝑣=1𝐴[𝑣]𝜉𝛾1𝐼𝑁+𝑚𝑣=1𝐴[𝑣]𝑇𝑖𝜉𝐼𝑠𝑚𝑣=1𝐴[𝑣]𝜉1̃𝑒1+𝛾2𝑚𝑣=1𝐴[𝑣]𝑇𝑖𝜉𝐼𝑠𝑚𝑣=1𝐴[𝑣]𝜉1𝑣𝛾3𝑣,𝑣𝐶𝑁𝑆,0,0,𝑖=0,1,2,,𝑠.(3.13)

Proof. This lemma can be proved in similar way as that of the one in [20, Lemmas 3.5–3.7].

Theorem 3.6. Suppose the ARKLM (2.1a), (2.1b), and (2.1c) is DA, DAS, and ASI stable, then there exist positive constants 0, 𝛾3, 𝛾4, 𝛾5, which depend only on the method and the parameters 𝜎𝜈, 𝑟𝜈, such that for (0,0]𝜀𝑖(𝑛)1+𝛾4𝜀max0(𝑛+1),max𝑖,𝑝𝑣𝜀(𝑛1𝑚𝑣+𝑝𝑣)𝑖+𝛾5max1𝑖𝑠𝜌𝑖(𝑛1)+𝑄𝑛1+𝛾3𝛾𝑛1,𝑖=0,1+𝛾4𝜀max0(𝑛+1),max𝑖,𝑝𝑣𝜀(𝑛𝑚𝑣+𝑝𝑣)𝑖+𝛾5max1𝑖𝑠𝜌𝑖(𝑛)+𝑄𝑛+𝛾3𝛾𝑛,𝑖=1,2,,𝑠,(3.14) where 𝜀0(𝑛)=̆𝑦𝑛𝑦𝑛,𝜀𝑖(𝑛)=̆𝑦𝑖(𝑛)𝑦𝑖(𝑛),𝐸=𝑖,𝑝𝑣,1𝑖𝑠,𝑑𝑝𝑣𝛾,𝐸𝑚={(𝑗,𝑣)0𝑗𝑠,1𝑣𝑚}.(3.15)

Proof. Using (3.10) and Lemma 3.5, for (0,0], we obtain that 𝜀0(𝑛+1)1+𝛾2𝜀0(𝑛)+𝛾3𝑚𝑣=1𝐴[𝑣]𝑓[𝑣]𝑡𝑛,𝑦(𝑛),̆̃𝑦[𝑣](𝑛)𝑓[𝑣]𝑡𝑛,𝑦(𝑛),̃𝑦[𝑣](𝑛)+𝑚𝑣=1̃𝑏[𝑣]𝑇𝑓[𝑣]𝑡𝑛,𝑦(𝑛),̆̃𝑦[𝑣](𝑛)𝑓[𝑣]𝑡𝑛,𝑦(𝑛),̃𝑦[𝑣](𝑛)+𝛾3𝑟𝑛+𝑄𝑛1+𝛾2𝜀0(𝑛)+𝛾3𝑟𝑛+𝑄𝑛+𝛾3𝑚𝑣=1𝑠𝑖=1𝑠𝑗=1𝑎[𝑣]𝑖𝑗𝑓[𝑣]𝑡𝑛,𝑐𝑗,𝑦𝑗(𝑛),̆̃𝑦𝑗[𝑣](𝑛)𝑓[𝑣]𝑡𝑛,𝑐𝑗,𝑦𝑗(𝑛),̃𝑦𝑗[𝑣](𝑛)2+𝑚𝑣=1𝑠𝑗=1𝑏𝑗[𝑣]𝑓[𝑣]𝑡𝑛+𝑐𝑗,𝑦𝑗(𝑛),̆̃𝑦𝑗[𝑣](𝑛)𝑓[𝑣]𝑡𝑛+𝑐𝑗,𝑦𝑗(𝑛),̃𝑦[𝑣](𝑛)1+𝛾2𝜀0(𝑛)+𝛾3𝛾𝑛+𝑄𝑛+𝛾3𝑚𝑣=1𝑠𝑖=1𝑠𝑗=1||𝑎[𝑣]𝑖𝑗||𝛾𝑣̆̃𝑦𝑗[𝑣](𝑛)̃𝑦𝑗[𝑣](𝑛)2+𝑚𝑠𝑣=1𝑗=1||𝑏𝑗(𝑣)||𝛾𝑣̆̃𝑦𝑗[𝑣](𝑛)̃𝑦𝑗[𝑣](𝑛)1+𝛾2𝜀0(𝑛)+𝛾3𝛾𝑛+𝑄𝑛+𝑚𝑣=1𝛾𝑣𝛾3𝑠𝑖=1𝑠𝑗=1||𝑎[𝑣]𝑖𝑗||2+𝑠𝑗=1||𝑏𝑗[𝑣]||max(𝑗,𝑣)𝐸𝑚̆̃𝑦𝑗[𝑣](𝑛)̃𝑦𝑗[𝑣](𝑛).(3.16) Moreover, it follows from (2.7) and (3.6c) that ̆̃𝑦𝑗[𝑣](𝑛)̃𝑦𝑗[𝑣](𝑛)sup𝛿𝑣[𝑟0,1)𝑃𝑣=𝑑||𝐿𝑃𝑣𝛿𝑣||max𝑑𝑃𝑣𝑟𝜀(𝑛𝑚𝑣+𝑃𝑣)𝑗+𝜌𝑗[𝑣](𝑛).(3.17) Substituting (3.17) in (3.16), we get 𝜀0(𝑛+1)1+𝛾4(0)max𝑗,𝑃𝑣𝐸𝜀0(𝑛)𝜀,max(𝑛𝑚𝑣+𝑃𝑣)𝑗+𝛾5(0)max(𝑗,𝑣)𝐸𝑚𝜌𝑗[𝑣](𝑛)+𝑄𝑛+𝛾3𝑟𝑛0,0,(3.18) where 𝛾4(0)=𝛾2+𝛾𝑠(0)sup𝛿𝑣[0,1)𝑟𝑃𝑣=𝑑|𝐿𝑃𝑣(𝛿𝑣)|, 𝛾5(0)=𝑚𝑣=1𝛾𝑣(𝛾3𝑠𝑖=1(𝑠𝑗=1|𝑎[𝑣]𝑖𝑗|)2+𝑠𝑗=1|𝑏𝑗[𝑣]|).
By Lemma 3.5, similar to (3.18), the inequalities 𝜀𝑖(𝑛)1+𝛾4(𝑖)max(𝑗,𝑃𝑣)𝐸𝜀0(𝑛),max(𝑗,𝑃𝑣)𝐸𝜀(𝑛𝑚𝑣+𝑃𝑣)𝑗+𝛾5(𝑖)max(𝑗,𝑣)𝐸𝑚𝜌𝑗[𝑣](𝑛)+𝑄𝑛+𝛾3𝑟𝑛𝑖=1,2,,𝑠,0,0,(3.19) follow, where 𝛾4(𝑖)=𝛾2+𝛾𝑠(𝑖)sup𝛿𝑣𝑟[0,1)𝑃𝑣=𝑑||𝐿𝑃𝑣𝛿𝑣||,𝛾5(𝑖)=𝑚𝑣=1𝛾𝑣𝛾3𝑠𝑖=1𝑠𝑗=1||𝑎[𝑣]𝑖𝑗||2+𝑠𝑗=1||𝑎[𝑣]𝑖𝑗||.(3.20) Setting 𝛾4=max{𝛾4(𝑖)0𝑖𝑠}, 𝛾5=max{𝛾5(𝑖)0𝑖𝑠}, and combining (3.18) with (3.19), we immediately obtain the conclusion of this theorem.

Now, we turn to study the convergence of ARKLM (2.1a), (2.1b), and (2.1c) for (1.1). It is always assumed that the analytic solution 𝑦(𝑡) of (1.1) is smooth enough on each internal of the form (𝑡0+(𝑗1),𝑡0+𝑗) (𝑗 is a positive integer) as (3.2) defined.

Theorem 3.7. Assume ARKLM (2.1a), (2.1b), and (2.1c) with stage order 𝑝 is DA, DAS, and ASI stable, then the ARKLM (2.1a), (2.1b), and (2.1c) is D-Convergent of order min{𝑝,𝑞+1}, where 𝑞=𝑑+𝑟.

Proof. By Theorem 3.6, we have for (0,0]𝜀𝑖(𝑛)1+𝛾4𝜀max0(𝑛1),max(𝑖,𝑃𝑣)𝐸𝜀(𝑛1𝑚𝑣+𝑃𝑣)𝑖+𝑇1𝑃+1+𝑇2𝑞+2,𝑖=0,1+𝛾4𝜀max0(𝑛),max𝑖,𝑃𝑣𝐸𝜀(𝑛𝑚𝑣+𝑃𝑣)𝑖+𝑇1𝑃+1+𝑇2𝑞+2,𝑖=1,2,,𝑠,(3.21) where 𝑇1=𝑀0+𝛾3𝑠𝑖=1𝑀2𝑖,𝑇2=𝛾5(𝑞+1)!𝑚𝑟𝑣=1𝑃𝑣=𝑑||𝛿𝑣𝑃𝑣||𝑀𝑞+1.(3.22) It follows from an induction to (3.21) for 𝑛 that 𝜀𝑖(𝑛)𝑛𝑗=01+𝛾4𝑗𝑇1𝑝+1+𝑇2𝑞+2,𝑖=0,𝑛+1𝑗=01+𝛾4𝑗𝑇1𝑝+1+𝑇2𝑞+2,𝑖=1,2,,𝑠,0,0.(3.23) Hence, for (0,0], we arrive at 𝑦𝑡𝑛𝑦𝑛=𝜀0(𝑛)𝑛𝑗=01+𝛾4𝑗𝑇1𝑝+1+𝑇2𝑞+2=1+𝛾4𝑛+11𝛾4𝑇1𝑝+1+𝑇2𝑞+2exp(𝑛+1)𝛾41𝛾4𝑇1𝑝+𝑇2𝑞+1𝑐(𝑡)min{𝑝,𝑞+1},(3.24) where 𝑐(𝑡)=exp𝑡𝑡0𝛾4exp0𝛾41𝛾4𝑇1+𝑇20𝑞+1𝑝,𝑝𝑞,exp𝑡𝑡0𝛾4exp0𝛾41𝛾4𝑇10𝑝𝑞1+𝑇2,𝑝>𝑞.(3.25) Therefore, the ARKLM (2.1a), (2.1b), and (2.1c) is D-Convergent of order min{𝑝,𝑞+1}, (𝑞=𝑟+𝑑).

4. Some Examples

In this final section we give some ARK methods to illustrate our theory in this paper.

Example 4.1. The two-stage additive RK method 012121001212011212012(4.1) with order one is GDN stable by Theorem 2.4, since 𝑀11=14141414,𝑀12=01002,𝑀21=01002,𝑀22=03004(4.2) are nonnegative definite.

Moreover, the method (4.1) is also DA, ASI, and DAS stable, since𝐼22𝑉=1𝐴[𝑉]1𝜉=12𝜉12𝜉123𝜉2𝜉isregularfor𝜉𝐶={𝜉𝐶Re𝜉0},(4.3a)𝐼22𝑉=1𝐴[𝑉]𝜉=(3/2)𝜉(1/2)𝜉(1/2)𝜉1(1/2)𝜉𝜉2(3/2)𝜉,(4.3b)2𝑉=1𝐴[𝑉]𝑇1𝜉𝐼22𝑉=1𝐴[𝑉]𝜉1=(1/2)𝜉2,(1/2)𝜉2+(1/2)𝜉𝜉2(3/2)𝜉,(4.3c)2𝑉=1𝐴[𝑉]𝑇2𝜉𝐼22𝑉=1𝐴[𝑉]𝜉1=(3/2)𝜉2,(3/2)𝜉(1/2)𝜉2𝜉2(3/2)𝜉,(4.3d) and (4.1), (4.3a)–(4.3d) are uniformly bounded for 𝜉C, 𝑅1(𝜉)=1+2𝑉=1𝐴[𝑉]𝑇1𝜉𝐼22𝑉=1𝐴[𝑉]𝜉1𝑒=1+(1/2)𝜉𝜉2=𝜉(3/2)𝜉2𝜉𝜉2(3/2)𝜉,(4.3e)𝑅2(𝜉)=1+2𝑉=1𝐴[𝑉]𝑇2𝜉𝐼22𝑉=1𝐴[𝑉]𝜉1𝑒=𝜉2+(3/2)𝜉𝜉2(3/2)𝜉=1,(4.3f) and (4.3e)-(4.3f) satisfy that |𝑅𝑖(𝜉)|1 for 𝜉𝐶, 𝑖=1,2.

By Theorem 3.7, we know that the ARKLM (2.1a), (2.1b), and (2.1c) corresponding to the method (4.1) is D-Convergent of order one.

Example 4.2. The two-stage additive RK method 110110121201121201(4.4) is strongly algebraically stable, since 𝑀11=340014,𝑀12=120012,𝑀21=120012,𝑀22=0001(4.5) are nonnegative definite.

Moreover, the method (4.4) is also DA, ASI, and DAS stable. Since𝐼22𝑉=1𝐴[𝑉]1𝜉=12𝜉023𝜉12𝜉isregularfor𝜉𝐶={𝜉𝐶Re𝜉0},(4.6a)𝐼22𝑉=1𝐴[𝑉]𝜉=1(3/2)𝜉(1/2)𝜉012𝜉1(1/2)𝜉+3𝜉2,(4.6b)2𝑉=1𝐴[𝑉]𝑇1𝜉𝐼22𝑉=1𝐴[𝑉]𝜉1=2𝜉3𝜉2,𝜉21(7/2)𝜉+3𝜉2,(4.6c)2𝑉=1𝐴[𝑉]𝑇2𝜉𝐼22𝑉=1𝐴[𝑉]𝜉1=(1/2)𝜉(3/4)𝜉2,(3/2)𝜉(11/4)𝜉21(7/2)𝜉+3𝜉2,(4.6d) and (4.6b)-(4.6d) are uniformly bounded for 𝜉𝐶, 𝑅1(𝜉)=1+2𝑉=1𝐴[𝑉]𝑇1𝜉𝐼22𝑉=1𝐴[𝑉]𝜉1𝑒=1(3/2)𝜉+𝜉21(7/2)𝜉+3𝜉2,(4.6e)𝑅2(𝜉)=1+2𝑉=1𝐴[𝑉]𝑇2𝜉𝐼22𝑉=1𝐴[𝑉]𝜉1𝑒=1(3/2)𝜉(1/2)𝜉21(7/2)𝜉+3𝜉2,(4.6f) and (4.6e) and (4.6f) satisfy that |𝑅𝑖(𝜉)|1, for 𝜉𝐶, 𝑖=1,2.

By Theorems 2.4 and 3.7 we know that the ARKLM (2.1a), (2.1b), and (2.1c) corresponding to the method (4.4) is GDN stable and D-Convergent of order two.


This paper was supported by the National Natural Science Foundation of China (11101109) and the Natural Science Foundationof Hei-long-jiang Province of China (A201107).


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