1. Introduction
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., [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:
()
()
where
τ1 ≤
τ2 ≤ ⋯≤
τm =
τ,
f[v] : [
t0,
T] ×
CN ×
CN →
CN,
v = 1,2, …,
m, and
φ,
ψ : [
t0 −
τ,
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 that
()
for
all
t ∈ [
t0,
T],
for
all
y,
y1,
y2,
u,
u1,
u2 ∈
CN, where
σv,
rv 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 [
7], 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 [
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 + h), that is,
()
Here the coefficients
,
, and
cj satisfy
()
tn =
t0 +
nh,
yn,
,
are approximations to the analytic solution
y(
tn),
y(
tn +
cjh),
y(
tn +
cjh −
τv) of (
1.1), respectively, and the argument
is determined by
()
With
τv = (
mv −
δv),
hδv ∈ [0,1), integer
mv ≥
r + 1,
r,
d ≥ 0, and
()
We assume
mv ≥
r + 1 is to guarantee that no (unknown) values
with
i ≥
n are used in the interpolation procedure. In addition, we always put
whenever
n < 0, and
yn =
φ(
tn) whenever
n ≤ 0.
The coefficients of the method may be organized in the Butcher tableau
()
where
C = [
c1,
c2, …,
cs]
T and for
v = 1,2, …,
m,
()
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)
T ∈
RS,
is the Kronecker product of matrix
G and
IN. For
u = (
u1,
u2, …,
us)
T,
v = (
v1,
v2, …,
vs)
T ∈
CNS, we define the inner product and the induced norm in
CNS 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 [
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
()
where constant
C > 0 depends only on the method, the parameter
σv,
v = 1,2, …,
m, and the interval length
T −
t0.
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
()
for
μ,
γ = 1,2, …,
m.
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
h, 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
()
where
.
Proof. From (2.11) we get
()
If the matrices
Mγμ are nonnegative definite, then
()
Furthermore, by conditions (
1.3) and (
1.4) and Schwartz inequality we have
()
From (
1.4), we know 0 ≤
rv ≤ −
σv.
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
E = {(
j,
Pv)
1 ≤
j ≤
s,
−
d ≤
Pv ≤
r}.
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
mv ≥
r + 1 that
()
Suppose for
n ≤
k (
k ≥ 0) that
()
Then from (
2.19), (
2.20),
mv ≥
r + 1, and
, we conclude that
()
This completes the proof of inequalities (
2.21). In view of (
2.21), we get for
n ≥ 0 that
()
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 by
()
where
()
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],
Qn,
rn, and
ρ[v](n) can be determined, respectively, as following:
()
where
q =
d +
r,
, and
satisfy
,
i = 0,1, 2, …,
s,
h ∈ (0,
h0],
h0 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
,
i = 1,2, …,
s,
f2(
t,
u,
v) is the Jacobian matrix (
∂f(
t,
u,
v)/
∂u)(
t ∈
R,
u,
v ∈
CN).
Assume that
) is regular, from (
3.9), we can get
()
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) (n ≤ 0), and 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
()
where the maximum stepsize
h0 depends on characteristic parameter
σv and the method, the function
C(
t) depends only on some
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 is regular for ξ ∈ C−∶ = {ξ ∈ C∣Reξ ≤ 0} and |Ri(ξ)| ≤ 1, for all ξ ∈ C−, i = 0,1, …, s, 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 ξ ∈ C−, and is uniformly bounded for ξ ∈ C−.
Definition 3.4. The ARKLM (2.1a), (2.1b), and (2.1c) is said to be DAS stable if the matrix is regular for ξ ∈ C−, and (i = 0,1, …, s) is uniformly bounded for ξ ∈ C−.
Lemma 3.5. Suppose the ARKLM is DA, DAS, and ASI stable, then there exist positive constants h0, γ1, γ2, γ3, which depend only on the method and the parameter σν, rν, 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 h0, γ3, γ4, γ5, which depend only on the method and the parameters σν, rν, such that for h ∈ (0, h0]
()
where
()
Proof. Using (3.10) and Lemma 3.5, for h ∈ (0, h0], 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
()
follow, where
()
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 y(t) of (1.1) is smooth enough on each internal of the form (t0 + (j − 1)h, t0 + jh) (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 h ∈ (0, h0]
()
where
()
It follows from an induction to (
3.21) for
n that
()
Hence, for
h ∈ (0,
h0], we arrive at
()
where
()
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
()
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
ξ ∈ C
−,
()
()
and (
4.3e)-(
4.3f) satisfy that |
Ri(
ξ)| ≤ 1 for
ξ ∈
C−,
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
()
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
ξ ∈
C−,
()
()
and (
4.6e) and (
4.6f) satisfy that |
Ri(
ξ)| ≤ 1
, for
ξ ∈
C−,
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.
Acknowledgments
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).
