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.
Delay differential equations arise in a variety of fields as biology, economy, control theory, electrodynamics (see, e.g., [1–5]). 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., ). 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 , Bellen , and Zennaro [11, 12].
In this paper, we consider the following nonlinear DDEs with delays:
where , , , and 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 that, where , are constants with
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 , it is straightforward to prove that under the conditions (1.3) and (1.4), the analytic solutions satisfy
To demand the discrete numerical solutions to preserve the stability properties (1.5) of the analytic solutions, Torelli  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 ). A disappointing conclusion is, as it is described in , 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  relaxed the RN stability restriction, considered the GDN stability and D-Convergence of (1.1) in the case . In 2001, Zhang et al.  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 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 stages and levels for (1.1) is a one-step numerical method which the numerical solution of (1.1) from (numerical approximation at ) to (numerical approximation at ), that is,
Here the coefficients , , and satisfy
, , , are approximations to the analytic solution , , of (1.1), respectively, and the argument is determined by
With , , integer , , and
We assume is to guarantee that no (unknown) values with are used in the interpolation procedure. In addition, we always put whenever , and whenever .
The coefficients of the method may be organized in the Butcher tableau
where and for ,
In order to write (2.1a), (2.1b), and (2.1c) in a more compact way we introduce some notations. The identity matrix will be denoted by , , is the Kronecker product of matrix and . For ,, we define the inner product and the induced norm in as follows:
Moreover, we also adopt that
With the above notation, method (2.1a), (2.1b), and (2.1c) can be written as
In 1997, Zhang and Zhou  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 and to the solution of (1.1) and (1.2), respectively, satisfy
where constant depends only on the method, the parameter , , and the interval length .
Here, we can see the constant need not to be less than 1, otherwise the Definition 2.2 is just RN stable in .
Definition 2.3. An ARKLM (2.1a), (2.1b), and (2.1c) is called strongly algebraically stable if matrices are nonnegative definite, where
Let and 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 write
Then (2.1a) reads
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
Proof. From (2.11) we get
If the matrices are nonnegative definite, then
Furthermore, by conditions (1.3) and (1.4) and Schwartz inequality we have
From (1.4), we know . Then, we have
Substituting (2.16) into (2.14), yields
In addition, with (2.1c), we have
Combining (2.17) with (2.18) and using (2.1b) we arrive at
where . Similar to (2.19), the inequalities
follow. In the following, with the help of inequalities (2.19), (2.20) and induction we will prove the inequalities:
In fact, it is clear from (2.19), (2.20), and that
Suppose for that
Then from (2.19), (2.20), , and , we conclude that
This completes the proof of inequalities (2.21). In view of (2.21), we get for that
As a result, we know that method (2.1a), (2.1b), and (2.1c) is GDN stable.
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.  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 . 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 . We assume the analytic solution of (1.1) is smooth enough and its derivatives used later are bounded by
If we introduce some notations
with the above notations, the local errors in (2.7) can be defined as with
If we take and
Then we can get the perturbed scheme of (2.7) With perturbations
According to Taylor formula and the formula in [19, pages 205–212], , and can be determined, respectively, as following:
where , , and satisfy , , , depends only on the method, and depends only on the method and some in (3.2).
Combining (2.7) with (3.6a), (3.6b), and (3.6c) yields the following recursion scheme for the :
where , and , , is the Jacobian matrix .
Assume that ) is regular, from (3.9), we can get
Now, we introduce the concept of D-Convergence from .
Definition 3.1. An ARKLM (2.1a), (2.1b), and (2.1c) with , and is called D-Convergence of order if this method, when applied to any given DDEs (1.1) subject to (1.3) and (1.4), produces an approximation sequence , and the global error satisfies a bound of the form
where the maximum stepsize depends on characteristic parameter and the method, the function depends only on some in (3.2), delay , characteristic parameters , and the method, .
Definition 3.2. The ARKLM (2.1a), (2.1b), and (2.1c) is said to be DA stable if the matrix is regular for and , for all , , where
Definition 3.3. The ARKLM (2.1a), (2.1b), and (2.1c) is said to be ASI stable if the matrix is regular for , and is uniformly bounded for .
Definition 3.4. The ARKLM (2.1a), (2.1b), and (2.1c) is said to be DAS stable if the matrix is regular for , and ) is uniformly bounded for .
Lemma 3.5. Suppose the ARKLM is DA, DAS, and ASI stable, then there exist positive constants , , , , which depend only on the method and the parameter , , such that
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 , , , , which depend only on the method and the parameters , , such that for
Proof. Using (3.10) and Lemma 3.5, for , we obtain that
Moreover, it follows from (2.7) and (3.6c) that
Substituting (3.17) in (3.16), we get
where , . By Lemma 3.5, similar to (3.18), the inequalities
Setting , , 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 of (1.1) is smooth enough on each internal of the form ( is a positive integer) as (3.2) defined.
Theorem 3.7. Assume ARKLM (2.1a), (2.1b), and (2.1c) with stage order is DA, DAS, and ASI stable, then the ARKLM (2.1a), (2.1b), and (2.1c) is D-Convergent of order , where .
Proof. By Theorem 3.6, we have for
It follows from an induction to (3.21) for that
Hence, for , we arrive at
Therefore, the ARKLM (2.1a), (2.1b), and (2.1c) is D-Convergent of order , .
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
with order one is GDN stable by Theorem 2.4, since
are nonnegative definite.
Moreover, the method (4.1) is also DA, ASI, and DAS stable, since
and (4.1), (4.3a)–(4.3d) are uniformly bounded for ,
and (4.3e)-(4.3f) satisfy that for , .
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
is strongly algebraically stable, since
are nonnegative definite.
Moreover, the method (4.4) is also DA, ASI, and DAS stable. Since
and (4.6b)-(4.6d) are uniformly bounded for , and (4.6e) and (4.6f) satisfy that , for , .
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).
T. E. Weldon, J. Kirk, and H. M. Finlay, “Cyclical granulopoiesis in chronic granulocytic leukemia: a simulation study,” Blood, vol. 43, pp. 379–387, 1974.
A. E. Ruehli, U. Miekkala, A. Bellen, and H. Heeb, “Stable time domain solutions for EMC problems using PEEC circuit models,” in Proceedings of the IEEE International Symposium on Electromagnetic Compatibility, 1994.
R. K. Brayton, “Small signal stability criterion for networks containing lossless transmission lines,” IBM Journal of Research and Development, vol. 12, pp. 431–440, 1968.
C. J. Zhang and S. Z. Zhou, “Nonlinear stability and D-convergence of Runge-Kutta methods for delay differential equations,” Journal of Computational and Applied Mathematics, vol. 85, no. 2, pp. 225–237, 1997.View at Publisher · View at Google Scholar