Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol103.Issue2:

Journal of Computational and Applied Mathematics 103 (1999) 263–279

Stability and error analysis of one-leg
methods for nonlinear delay dierential equations 1
Chengming Huanga; b; ∗ , Shoufu Lia , Hongyuan Fub , Guangnan Chenb
a

Institute for Computational and Applied Mathematics, Xiangtan University, Hunan,
411105, People’s Republic of China
b
Graduate School, CAEP, P.O. Box 2101, Beijing, 100088, People’s Republic of China
Received 2 July 1998; received in revised form 7 November 1998

Abstract
This paper is concerned with the numerical solution of delay dierential equations (DDEs). We focus on the stability
behaviour and error analysis of one-leg methods with respect to nonlinear DDEs. The new concepts of GR-stability,
GAR-stability and weak GAR-stability are introduced. It is proved that a strongly A-stable one-leg method with linear
interpolation is GAR-stable, and that an A-stable one-leg method with linear interpolation is GR-stable, weakly GAR-stable
c 1999 Elsevier Science B.V. All rights
and D-convergent of order s, if it is consistent of order s in the classical sense.
reserved.

AMS classication: 65L06; 65L20
Keywords: Nonlinear delay dierential equations; One-leg methods; Stability; D-convergence

1. Introduction
In recent years, many papers discussed numerical methods for the solution of delay dierential
equations (DDEs) (see [1, 2, 9, 12–14, 19–24] and their references). They mainly focused on the
stability of numerical methods for linear scalar model equation
y′ (t) = ay(t) + by(t−);
y(t) = (t);

∗
1

t¿0;

t 6 0;

Corresponding author. E-mail: cmhuang@xtu.edu.cn.
Project supported by NSF of China.

c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 6 2 - 3

(1.1)

264

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

where a; b are complex numbers which satisfy
|b|¡−Re(a);
(¿0) is constant delay, and (t) is a continuous function. The concepts of P-stability and GPstability were introduced and a signicant number of results have already been found for both
Runge–Kutta methods and linear multistep methods. Recently, the stability of RK methods has been
studied in [13] based on the following test problem:
y′ (t) = Ly(t) + My(t − );

t¿0;
(1.2)

y(t) = (t);

t 6 0;

where L; M denote constant, complex matrices. However, we can not say that a stable method for
(1.1) or (1.2) is also valid for a more general system of nonlinear DDEs. The stability results of
numerical methods for nonlinear DDEs are much less. Up to now, we only see the stability analysis
of some methods (cf. [2, 20, 23]). In this paper, we investigate the stability of one-leg methods with
respect to nonlinear DDEs.
On the other hand, error analysis of numerical methods for DDEs is mostly based on Lipschitz
conditions. For sti DDEs, however, the Lipschitz constant will be very large, so that the classical
convergence theory can not be applied. In this paper, in addition to stability analysis, we will also
investigate the error behaviour and obtain the global error bounds independent on the stiness of
the underlying system. We will continue to analyse stability and error behaviour of Runge–Kutta
methods for nonlinear DDEs in other papers.
This paper is structured as follows: In Section 2, we x our attention on a particular class of
DDEs, collecting several results from the literature. In Section 3, some new concepts of stability are
introduced for nonlinear DDEs. They are reminiscent of that for the sti ODEs eld. In Section 4, we
analyse the stability of A-stable one-leg methods with linear interpolation with respect to nonlinear
DDEs. In Section 5, we investigate the error behaviour of A-stable one-leg methods with respect to

nonlinear sti DDEs. In Section 6, we brie
y discuss the numerical solution of DDEs with several
delays.
2. Test problems
Let h·; ·i be an inner product on C N and k · k the corresponding norm. Consider the following
nonlinear equation:
y′ (t) = f(t; y(t); y(t − ));
y(t) = 1 (t);

t 6 0;

t ¿ 0;
(2.1)

where  is a positive delay term, 1 is a continuous function, and f : [0; +∞) × C N × C N → C N , is
a given mapping. In order to make the error analysis feasible, we always assume that the problem
(2.1) have a unique solution y(t) which is suciently dierentiable and satises

i
d y(t)

dt i
6 Mi :

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

265

Denition 2.1. Let p; q be real constants. The class of all problems (2.1) with f satisfying the
following conditions:
Rehu1 − u2 ; f(t; u1 ; v)−f(t; u2 ; v)i 6 pku1 − u2 k2 ;
kf(t; u; v1 ) − f(t; u; v2 )k 6 qkv1 − v2 k;

t ¿ 0; u1 ; u2 ; v ∈ C N ;

t ¿ 0; u; v1 ; v2 ∈ C N ;

(2.2)

(2.3)

is denoted by Dp; q .
Remark 2.2. In the literature with respect to nonlinear stability and B-convergence of numerical
methods for ODEs, the class Dp; 0 has been used widely as the test problem class (cf. [3, 4, 6, 7, 9,
18]).
Proposition 2.3. System (1.1) belongs to the class Dp; q where p = Re(a) and q = |b|.
Proposition 2.4. System (1.2) belongs to the class Dp; q where p = (L); (·) is the logarithmic
matrix norm corresponding to the inner-product norm on C N ; and q = supkxk=1 kMxk.
For the nonlinear case, consider the following example (cf. [8]):
y′ (t) = −ay(t) +

by(t − )
;
1 + [y(t − )]n

where a¿0 and b are real parameters and n is an even positive integer. This equation is a model for
respiratory diseases, where y(t) represents the concentration of carbon dioxide at time t. Obviously,
this equation belongs to the class Dp; q with p = −a and q = |b| for n = 2 or 4.
In order to discuss stability and asymptotic stability of DDEs (2.1) of the class Dp; q , we introduce

another system, dened by the same function f(t; u; v), but with another initial condition:
z ′ (t) = f(t; z(t); z(t − ));
z(t) = 2 (t);

t ¿ 0;

t 6 0:

(2.4)

Proposition 2.5. Suppose systems (2.1) and (2.4) belong to the class Dp; q with q 6 −p. Then the
following is true:
ky(t) − z(t)k 6 max k1 (x) − 2 (x)k;
x60

t ¿ 0:

(2.5)

The proof of this proposition can be found in [20].

Proposition 2.6. Suppose systems (2.1) and (2.4) belong to the class Dp; q with q ¡ −p. Then the
following holds:
lim ky(t) − z(t)k = 0:

t→+∞

The proof of this proposition can be found in [23], where a more general result was given.

(2.6)

266

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

3. Some concepts
We brie
y recall the form of a one-leg method for the numerical solution of the ordinary dierential equation
y′ (t) = f(t; y(t));

t ¿ 0;

y(0) = y0 :

(3.1)

The one-leg k step method is the following:
(E)yn = hf ((E)tn ; (E)yn );

(3.2)

where h¿0 is the step size, E is the translation operator:
Eyn = yn+1 , eachPyn is an approximation
P
to the exact solution y(tn ) with tn = nh, and (x) = kj=0 j x j and (x) = kj=0 j x j are generating
polynomials, which are assumed to have real cocients, no common divisor. We also assume (1)=
0; ′ (1) = (1) = 1.
Apply the one-leg k-step method (; ) to DDE (2.1)
(E)yn = hf((E)tn ; (E)yn ; y n );

n = 0; 1; 2; : : : ;

(3.3)

where the argument y n denotes an approximation to y((E)tn − ) that is obtained by a specic
interpolation at the point t = (E)tn −  using {yi }i6n+k .
Process (3.3) is dened completely by the one-leg method (; ) and the interpolation procedure
for y n .
It is well known that any A-stable one-leg method for ODEs has order at most 2. So we can use
the linear interpolation procedure for y n . Let  = (m − )h with integer m ¿ 1 and  ∈ [0; 1). We
dene
y n = (E)yn−m+1 + (1 − )(E)yn−m ;

(3.4)

where yl = 1 (lh) for l 6 0.
Similarly, apply the same method (; ) to DDE (2.4)
(E)zn = hf((E)tn ; (E)zn ; zn );

n = 0; 1; 2; : : : ;

(3.5)

zn = (E)zn−m+1 + (1 − )(E)zn−m ;

(3.6)

where zl = 2 (lh) for l 6 0.
Denition 3.1. A numerical method for DDEs is called R-stable if, under the condition that q 6−p,
there exists a constant C which depends only on the method,  and q, such that the numerical
approximations yn and zn to the solutions of any given problems (2.1) and (2.4) of the class Dp; q ,
respectively, satisfy the following inequality:
kyn − zn k 6 C



max kyj − zj k + max k1 (t) − 2 (t)k

06j6k−1

t60



(3.7)

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

267

for every n ¿ k and for every stepsize h ¿ 0 under the constraint
hm = ;

(3.8)

where m is a positive integer.
GR-stability is dened by dropping the restriction (3.8).
Remark 3.2. Torelli [20] introduced RN- and GRN-stability for numerical methods applied to nonautonomous nonlinear DDEs. They require that the dierence of two numerical solutions is bounded
by the maximum dierence of the initial values which means that the method is contractive. Here
we relax their requirements. R- and GR-stability only require the dierence to be controlled and
uniformly bounded. Therefore, R-stability is a weaker concept than RN-stability. In fact, if a method
is RN-stable, then we can choose C = 1 such that the method is R-stable.
Proposition 3.3. Any R-stable one-leg method is A-stable.
Denition 3.4. A numerical method for DDEs is called AR-stable if, under the condition that q ¡ −
p, the numerical approximations yn and zn to the solutions of any given problems (2.1) and (2.4)
of the class Dp; q , respectively, satisfy the condition
lim kyn − zn k = 0

n→∞

(3.9)

for every stepsize h ¿ 0 under the constraint (3.8).
GAR-stability is dened by dropping restriction (3.8).
Denition 3.5. A numerical method for DDEs is called weak AR-stable if, under the assumptions
of Denition 3.4, (3.9) holds when f further satises
kf(t; u1 ; v) − f(t; u2 ; v)k 6 Lku1 − u2 kb ; t ¿ 0; u1 ; u2 ; v ∈ C N ;

(3.10)

where b is a positive real number and L is a nonnegative real number.
Weak GAR-stability is dened by dropping the restriction (3.8).
Remark 3.6. If the function f(t; u; v) is uniformly Lipschitz continuous in variable u, then (3.10)
holds.
Remark 3.7. Both AR-stability and weak AR-stability can be regarded as the nonlinear analogues
of the concept of P-stability [1].
Up to now, error analyses of numerical methods for DDEs are mostly based on the function
f(t; u; v) satisfying Lipschitz conditions for u and v. For sti DDEs, however, the Lipschitz constant
with respect to u will be very large, so that the classical convergence theory cannot be applied.
Now, we introduce the concept of D-convergence for sti DDEs.

268

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

Denition 3.8. The one-leg method (3.3) with interpolation procedure (3.4) is said to be D-convergent of order s for the problem class Dp; q if this method when applied to any given problem
(2.1) of the class Dp; q with yi = y(ti ); i ¡ k, produces an approximation sequence {yn }, and the
global error satises a bound of the form
ky(tn ) − yn k 6 C(tn )h s ;

n ¿ k; h ∈ (0; h0 ];

where the function C(t) and the maximum stepsize h0 depend only on the method, some of the
bounds Mi , the delay , the characteristic parameters p and q of the problem class Dp; q .
Remark 3.9. D-convergence concept was rstly proposed by Zhang and Zhou [24] for Runge–Kutta
methods. It was assumed that q 6 − p in [24]. Here we drop this restriction. The D-convergence
concept is wider than the well-known B-convergence concept (see [6, 7, 9, 15, 16]). D-convergence
for the problem class Dp; 0 is just B-convergence. B-convergence results of one-leg methods can
be found in [11, 15].
4. Stability analysis
In this section, we focus on the stability analysis of A-stable one-leg methods with respect to the
nonlinear test problem class Dp; q .
Let yn ; zn ∈ C N ,





wn = 

yn − zn
yn+1 − zn+1
..
.



yn+k−1 − zn+k−1






and for a real symmetric positive denite k × k matrix G = [gij ], the norm k · kG is dened by


kU kG = 

k
X

i; j=1

1=2

gij hui ; uj i ;

U = (u1T ; u2T ; : : : ; ukT )T ∈ C kN :

Theorem 4.1. Assume that the one-leg k step method (; ) is G-stable for a real symmetric
positive denite matrix G and that the solved problems (2:1) and (2:4) belong to the class Dp; q ;
then
kwn+1 k2G 6 kw0 k2G + h

n
X

[(2p + q)k(E)(yj − zj )k2 + qk(E)(yj−m+1 − zj−m+1 )k2

j=0

+ (1 − )qk(E)(yj−m − zj−m )k2 ]:
Proof. Suppose the method is G-stable for G, then for all real a0 ; a1 ; : : : ; ak ,
AT1 GA1 − AT0 GA0 6 2(E)a0 (E)a0 ;

(4.1)

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

269

where Ai = (ai ; ai+1 ; : : : ; ai+k−1 )T ; i = 0; 1. Therefore, we can easily obtain (cf. [4, 15, 17])
kwn+1 k2G − kwn k2G 6 2Reh(E)(yn − zn ); (E)(yn − zn )i:

(4.2)

Hence
kwn+1 k2G − kwn k2G 6 2Reh(E)(yn − zn ); h(f((E)tn ; (E)yn ; y n )
−f((E)tn ; (E)zn ; zn ))i
6 2Reh(E)(yn − zn ); h(f((E)tn ; (E)yn ; y n ) − f((E)tn ; (E)zn ; y n ))i
+2Reh(E)(yn − zn ); h(f((E)tn ; (E)zn ; y n ) − f((E)tn ; (E)zn ; zn ))i
6 2h[pk(E)(yn − zn )k2 + qk(E)(yn − zn )k · ky n − zn k]
6 (2p + q)hk(E)(yn − zn )k2 + qhky n − zn k2 :
It follows from (3:4) and (3:6) that
ky n − zn k2 6 2 k(E)(yn−m+1 − zn−m+1 )k2 + (1 − )2 k(E)(yn−m − zn−m )k2
+ 2(1 − )k(E)(yn−m+1 − zn−m+1 )k · k(E)(yn−m − zn−m )k
6 2 k(E)(yn−m+1 − zn−m+1 )k + (1 − )2 k(E)(yn−m − zn−m )k
+ (1 − )(k(E)(yn−m+1 − zn−m+1 )k2 + k(E)(yn−m − zn−m )k2 )
= k(E)(yn−m+1 − zn−m+1 )k2 + (1 − )k(E)(yn−m − zn−m )k2 :
Therefore,
kwn+1 k2G − kwn k2G 6 (2p + q)hk(E)(yn − zn )k2 + qhk(E)(yn−m+1 − zn−m+1 )k2
+ (1 − )qhk(E)(yn−m − zn−m )k2 :
By induction we have that (4:1) holds, which completes the proof of Theorem 4.1.
Theorem 4.2. Any A-stable one-leg method (; ) is GR-stable.
Proof. Suppose the method is A-stable. Then k =k ¿ 0 and the method is G-stable (cf. [5]). Application of Theorem 4.1 in combination with the condition that 0 6 q 6 − p yields
kwn+1 k2G 6 kw0 k2G + ph

n
X

k(E)(yj − zj )k2 + qh

j=0

+ (1 − )qh

n−m
X

j=−m

n−m+1
X

j=−m+1

k(E)(yj − zj )k2 :

k(E)(yj − zj )k2

270

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

When m ¿ 2 we have
−1
X

kwn+1 k2G 6 kw0 k2G + qh

k(E)(yj − zj )k2 + (1 − )qh

+ (m − 1)qh

+ (1 − )mqh

max

= kw0 k2G + q

max

−m+16j6−1

−m6j6−1

6 kw0 k2G + (m − )qh

k(E)(yj − zj )k2

j=−m

j=−m+1

6 kw0 k2G

−1
X

k(E)(yj − zj )k2

max

−m6j6−1

max

−m6j6−1

k(E)(yj − zj )k

2

k(E)(yj − zj )k2

k(E)(yj − zj )k2 :

(4.4)

On the other hand, when m = 1 we have
kwn+1 k2G 6 kw0 k2G + (1 − )qhk(E)(y−1 − z−1 )k2
= kw0 k2G + qk(E)(y−1 − z−1 )k2 :

(4.5)

Combining (4:4) and (4:5) yields
kwn+1 k2G 6 kw0 k2G + q

max

−m6j6−1

k(E)(yj − zj )k2 ;

n ¿ 0; m ¿ 1:

(4.6)

Let 1 and 2 denote the maximum and minimum eigenvalue of the matrix G, respectively. Then
we have
2 kyn+k − zn+k k2 6 1

k−1
X

kyi − zi k2 + q

i=0

max

−m6j6−1

k(E)(yj − zj )k2 ;

n ¿ 0:

Hence,
kyn+k − zn+k k2 6

k1
q
max kyi − zi k2 +
max k(E)(yj − zj )k2 ;
2 06i6k−1
2 −m6j6−1

n ¿ 0:

(4.7)

This shows that the method is GR-stable.
In the following, we further investigate the asymptotic stability of one-leg methods. A method is
strongly A-stable if it is A-stable and the modulus of any root of (x) is strictly less than 1.
Theorem 4.3. A strongly A-stable one-leg method is GAR-stable.
Proof. Analogous to Theorem 4.2, we can easily obtain
kwn+1 k2G + (−p − q)h

n
X

k(E)(yj − zj )k2

j=0

6 kw0 k2G

+ q

max

−m6j6−1

k(E)(yj − zj )k2 ;

n ¿ 0:

(4.8)

Because −p − q ¿ 0 and h ¿ 0, we have
lim k(E)(yn − zn )k = 0:

n→∞

(4.9)

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

271

On the other hand,
(E)(yn − zn ) =

k
X

j (yn+j − zn+j );

(4.10)

j=0

which gives
wn+1 = Awn + Bn ;

(4.11)

where


0
0
..
.





A=

 0

 0

−

k

1
0
..
.

0
1
..
.

···
···
..
.

0
1
−
k

0
2
−
k

···
···

0
0
..
.

−







 ⊗IN ;

1 

k−1 

k







Bn = 


 1

k

0
0
..
.
0
(E)(yn − zn )







:




Here ⊗ is the Kronecker product, and IN is the N × N -identity matrices. From the strong A-stability
of the method, we have the spectral radius of the matrix A strictly less than 1. Therefore, there
exists a norm k · k∗ in C kN such that the corresponding operator norm kAk∗ = supkxk=1 kAxk∗ ¡ 1.
From kBn k∗ → 0 we know that, for any  ¿ 0, there exists l ¿ 0 such that kBn k∗ ¡ (1 − kAk∗ )=2
when n ¿ l. Hence,
kwn k∗ = kAn−l wl +

n−l−1
X

Aj Bn−j−1 k∗ 6 kAkn−l
∗ kwl k∗ + =2;

n ¿ l;

(4.12)

j=0

which shows that there exists N0 ¿ l such that kwn k ¡  when n ¿ N0 . Therefore, we have
lim kyn − zn k = 0;

n→∞

which completes the proof of Theorem 4.3.
Lemma 4.4. Suppose {i (x)}ri=1 are a basis of polynomials for P r−1 ; the space of polynomials of
degree strictly less than r; then there is always a unique solution yn ; : : : ; yn+r−1 to the system of
equations
i (E)yn = bi ;

bi ∈ C N ; i = 1; : : : ; r

(4.13)

and there exists a constant D; independent of the bi ; such that
max |yn+i | 6 D max |bi |:

06i6r−1

16i6r

(4.14)

Theorem 4.5. Any A-stable one-leg method (; ) is weak GAR-stable.
Proof. It follows from −p−q¿0 and h¿0 that
lim k(E)(yn − zn )k = 0:

n→∞

(4.15)

272

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

Considering (3.10), we further obtain
lim k(E)(yn − zn )k = 0:

(4.16)

n→∞

On the other hand,  and  are coprime, and both are of degree k. Hence, {xi (x); xi (x); i =
0; : : : ; k − 1} form a basis for P 2k−1 . Considering (4.15), (4.16) and Lemma 4.4, we have
lim kyn − zn k = 0;

n→∞

which completes the proof of Theorem 4.5.
Corollary 4.6. Any A-stable one-leg method (; ) is GP-stable.
5. Error analysis of A-stable one-leg methods
In this section, we focus on the error analysis of A-stable one-leg methods applied to sti DDEs.
For the sake of simplicity, we always assume that all constants hi ; ci and di used later are dependent
only on the method, some of the bounds Mi , the characteristic parameters p and q of the problem
class Dp; q , and .
Now, we consider scheme (3.3) and the following scheme:
(E)yˆn + k en = hf((E)tn ; (E)yˆn + k en ; Yn );

n = 0; 1; 2; : : : ;

(5.1)

where
c1 = −

k−1 
1X

2

j=0

j −

k
k X

k
j j 2 −
k
k

yˆn = y(tn ) + c1 h2 y′′ (tn );



j=0


2
k
X
1
jj +  jj  ;

2

(5.2a)

j=0

(5.2b)

Yn = y((E)tn − );

(5.2c)

where y(t) is the exact solution of problem (2.1). Then for any n¿0; en is uniquely determined by
Eq. (5.1).
Theorem 5.1. Assume that the one-leg method (; ) is G-stable with respect to G and that the
function f(t; u; v) satises conditions (2:2) and (2:3). Then there exist constant h1 ; d1 ; d2 and d3
such that
kn+1 k2G 6 (1 + d1 h)kn k2G + d2 hkyn − Yn k2 + d3 h−1 ken k2 ;

n = 0; 1; 2; : : : ;

(5.3)

where n = ((yn − yˆn )T ; (yn+1 − yˆn+1 )T ; : : : ; (yn+k−1 − yˆn+k−1 )T )T .
Proof. In view of G-stability, analogous to Theorem 4.1, we have
kn+1 k2G 6 kn k2G + 2Reh(E)(yn − yˆn ) − k en ; (E)(yn − yˆn ) − k en i;

(5.4)

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

273

where
n+1 = ((yn+1 − yˆn+1 )T ; (yn+2 − yˆn+2 )T ; : : : ; (yn+k − yˆn+k − en )T )T :
It follows from conditions (2.2) and (2.3) that
kn+1 k2G 6 kn k2G + 2hReh(E)(yn − yˆn ) − k en ; f((E)tn ; (E)yn ; yn )
−f((E)tn ; (E)yˆn + k en ; Yn )i
6 kn k2G + 2h[pk(E)(yn − yˆn ) − k en k2 + qk(E)(yn − yˆn ) − k en k · kyn − Yn k]
6 kn k2G + h[(2p + q)k(E)(yn − yˆn ) − k en k2 + qkyn − Yn k2 ]:
(5.5)
Let
c2 =



0;
2p + q 6 0;
2p + q; 2p + q¿0;

(5.6)

then
kn+1 k2G 6 kn k2G + h[c2 k(E)(yn − yˆn ) − k en k2 + qkyn − Yn k2 ]:

(5.7)

On the other hand, it follows from (3.3) and (5.1)
(E)(yn − yˆn ) − k en =

k−1 
X
j=0

+

k
j − j (yn+j − yˆn+j )
k


k
h( f((E)tn ; (E)yn ; yn ) − f((E)tn ; (E)yˆn + k en ; Yn )):
k

From G-stability of the method we have k =k ¿0 (cf. [4, 5]). Then
k(E)(yn − yˆn ) − k en k2
*

=Re (E)(yn − yˆn ) − k en ;

k−1 
X
j=0

k
j − j (yn+j − yˆn+j )
k


+

k
+ hReh(E)(yn − yˆn ) − k en ; f((E)tn ; (E)yn ; yn )
k
−f((E)tn ; (E)yˆn + k en ; Yn )i




k−1 


X
k


6 k(E)(yn − yˆn ) − k en k
j − j (yn+j − yˆn+j )

k

j=0


+

k
k
hpk(E)(yn − yˆn ) − k en k + hqkyn − Yn k :
k
k

When hp k =k ¡1, we have

k(E)(yn − yˆn ) − k en k 6

k
k
c3 kn kG + hqkyn − Yn k ;
k − k hp
k




(5.8)

274

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

where



k  p

c3 = max j − j 
2
06j6k−1
k

(5.9)

and 2 denotes the minimum eigenvalue of the matrix G. Therefore, together with (5.7) and (5.8)
kn+1 k2G

6 kn k2G

"

2c2 k2 h
c2 kn k2G +
+
(k − k hp)2 3

2c2 c32 k2 h
6 1+
(k − k hp)2

!

kn k2G



k
hq
k

2

kyn − Yn k

2c2 (k hq)2
+ q+
(k − k hp)2

2

!

#

+ qhkyn − Yn k2

hkyn − Yn k2 ;

k hp=k ¡1:

(5.10)

Let 1 denote the maximum eigenvalue of the matrix G, then
p

kn+1 k2G 6 kn+1 k2G + 1 ken k2 + 2 1 ken k · kn+1 kG


1
2
6 (1 + h)kn+1 kG + 1 +
1 ken k2 :
h

(5.11)

Let
h1 =

(

1; p 6 0;
min(1; 2kk p );

d1 = sup
h∈(0; h1 ]

d2 = sup
h∈(0; h1 ]

(5.12)

p¿0;
!

(5.13a)

(1 + h);

(5.13b)

2c2 c32 k2 (1 + h)
1+
;
(k − k hp)2
2c2 (k hq)2
q+
(k − k hp)2

!

d3 = max (1 + h)1 ;

(5.13c)

h∈(0; h1 ]

then (5.3) holds, which completes the proof of Theorem 5.1.
Theorem 5.2. Assume that the one-leg method (; ) applied to problem (2:1) of the class Dp; q
is consistent of order s 6 2 in the classical sense for ODEs and that k =k ¿0; then there exists
constant d4 such that
ken k 6 d4 hs+1 ;

h ∈ (0; h1 ];

n = 0; 1; 2; : : : ;

(5.14)

where h1 is dened by (5:12).
Proof. Consider the following scheme:
y((E)tn ) =

k−1 
X
j=0

k
k
j − j yˆn+j + hy′ ((E)tn ) + R1(n) ;
k
k


(E)yˆn = hy′ ((E)tn ) + R2(n) :

(5.15)
(5.16)

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

275

By Taylor expansion, there exists constant c4 such that
kR1(n) k 6 c4 h3 ;

(5.17)

kR2(n) k 6 c4 hs+1 :

(5.18)

From (5.1) and (5.15) we have
y((E)tn ) − (E)yˆn − k en =

k
h[f((E)tn ; y((E)tn ); y((E)tn − ))
k
− f((E)tn ; (E)yˆn + k en ; Yn )] + R1(n) :

(5.19)

In view of (2.2) we can obtain further
ky((E)tn ) − (E)yˆn − k en k2 6

k
hpky((E)tn ) − (E)yˆn − k en k2
k
+ kR1(n) k · ky((E)tn ) − (E)yˆn − k en k;

then
ky((E)tn ) − (E)yˆn − k en k 6 2kR1(n) k;

h ∈ (0; h1 ];

(5.20)

where h1 is dened by (5.12).
Substituting (5.19) with (5.20), we have
3k (n)
kh[y′ ((E)tn ) − f((E)tn ; (E)yˆn + k en ; Yn )]k 6
kR1 k;
k

h ∈ (0; h1 ]:

(5.21)

On the other hand, a combination of (5.1) and (5.16) yields
k en = hf((E)tn ; (E)yˆn + k en ; Yn ) − hy′ ((E)tn ) − R2(n) :
Hence
ken k 6

3
1
kR1(n) k +
kR(n) k;
|k |
|k | 2

h ∈ (0; h1 ]:

(5.22)

In view of (5.17) and (5.18), (5.14) holds, which completes the proof of Theorem 5.2.
Theorem 5.3. If an A-stable one-leg k step method (; ) is consistent of order s in the classical
sense for ODEs; then it is D-convergent of order s; where k ¿ 1; s = 1; 2.
Proof. Suppose the method (; ) is A-stable. Then k =k ¿0 and the method is G-stable. From
Theorems 5.1 and 5.2, we have
kn+1 k2G 6 (1 + d1 h)kn k2G + d2 hkyn − Yn k2 + d3 d4 h2s+1 ;

h ∈ (0; h1 ]:

By induction, it is easily seen that
kn+1 k2G 6 k0 k2G + h

n
X
i=0

[d2 kyi − Yi k2 + d1 ki k2G + d3 d4 h2s ];

h ∈ (0; h1 ]:

276

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

Therefore,
k−1
n
h X
1 X
kyj − yˆj k2 +
[d2 ky i − Yi k2
2 j=0
2 i=0

kyn+k − yˆn+k k2 6

+ d 1 1

k−1
X

kyi+j − yˆ i+j k2 + d3 d4 h2s ];

h ∈ (0; h1 ]:

(5.23)

j=0

On the other hand,
kyi − Yi k = ky i − y((E)ti − )k
6 k(E)(yi+1−m − yˆi+1−m )k + (1 − )k(E)(yi−m − yˆi−m )k
+ k(E)yˆi+1−m + (1 − )(E)yˆ i−m − y((E)ti − )k
6

k
X

|j |·kyi+j+1−m − yˆ i+j+1−m k + (1 − )

j=0

+ k

k
X

|j |·kyi+j−m − yˆ i+j−m k

j=0

k
X

j yˆ i+j+1−m + (1 − )

j=0

k
X

j yˆ i+j−m − y((E)ti − )k:

j=0

By Taylor expansion, there exist constants h2 and d5 such that


k+1
X
kyn − Yn k2 6 d5  kyi+j−m − yˆ i+j−m k2 + h4  ;

h ∈ (0; h2 ]:

(5.24)

j=0

Substituting (5.23) with (5.24), (5.2) and yj = y(tj ); j 6 k − 1, leads to
kyn+k − yˆ n+k k2 6

n+k−1
X
 1 2 2 4 1
kc1 M2 h + d1 kh
kyi − yˆ i k2
2
2
i=0









n
k+1
X
h X


d2 d5
+
kyi+j−m − yˆ i+j−m k2 + h4  + d3 d4 h2s 
2 i=0
j=0

"

n+k−1
X
1
6
(1 kc12 M22 + d2 d5 nh)h4 + nhd3 d4 h2s + 1 d1 kh
kyi − yˆ i k2
2
i=0

+ d2 d5 (k + 2)h

n+k+1−m
X

2

kyi − yˆ i k

i=−m

#

;

h ∈ (0; h3 ];

(5.25)

where h3 = min(h1 ; h2 ) 6 1; m ¿ 1; n ¿ 0; s = 1; 2:
Therefore, whether m = 1 or m¿1, it is certain that there exist constants h0 ; c0 and d0 such that
kyn+k − yˆn+k k2 6 c0 (1 + tn+k )h2s + hd0

n+k−1
X
i=0

kyi − yˆ i k2 ; n = 0; 1; 2; : : : ;

h ∈ (0; h0 ]:

(5.26)

277

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

By induction we can obtain
kyn+k − yˆ n+k k2 6 exp(d0 tn+k )(c0 (1 + tn+k )h2s + hd0 k max kyi − yˆ i k2 ); n = 0; 1; 2; : : : ;
06j¡k

h ∈ (0; h0 ]:

(5.27)

In fact (5.27) obviously holds when n = 0. If (5.27) holds for n ¡ l, where l is a positive integer,
then it follows from (5.26) that
kyl+k − yˆl+k k2 6 c0 (1 + tl+k )h2s + hd0

l+k−1
X

kyi − yˆ i k2

i=0

6 c0 (1 + tl+k )h2s + hd0 k max kyi − yˆ i k2 + hd0
06j¡k

l+k−1
X

[exp(d0 ti )(c0 (1 + ti )h2s

i=k

2

+ hd0 k max kyi − yˆ i k )]
06j¡k

2s

2

6 [c0 (1 + tl+k )h + hd0 k max kyi − yˆ i k ] 1 + hd0
06j¡k

l+k−1
X

exp(d0 ti )

i=k

!

6 exp(d0 tl+k )[c0 (1 + tl+k )h2s + hd0 k max kyi − yˆ i k2 ]:
06j¡k

This shows that (5.27) holds for every n ¿ 0. From (5.2) we further obtain
p

kyn+k − y(tn+k )k 6 |c1 |M2 h2 + exp( 21 d0 tn+k )[hs c0 (1 + tn+k ) +
n = 0; 1; 2; : : : ;

h ∈ (0; h0 ]:

p

d0 kh0 |c1 |M2 h2 ];
(5.28)

This shows that the method is D-convergent of order s; s = 1; 2, which completes the proof of
Theorem 5.3.
Now, we review results from the literature for one-leg methods. For ODEs, Dahlquist [5] proved
that A-stability is equivalent to G-stability. Li [15] proved that A-stability implies B-convergence,
and Huang [10] further proved that B-convergence implies A-stability. For DDEs, from Denition 3.1
and Theorem 4.2, A-stability and GR-stability are equivalent. From Denition 3.8, D-convergence
implies B-convergence. Theorem 5.3 shows that A-stability implies D-convergence. Therefore we
have the following result.
Theorem 5.4. For a one-leg k-step method (3.3) with linear interpolation (3.4), the following
statements are equivalent:
(1) (; ) is A-stable.
(2) (; ) is G-stable.
(3) (; ) is B-convergent.
(4) (; ) is GR-stable.
(5) (; ) is D-convergent.
For solvability of Eq. (3.3) with (3.4), we refer to [4] when m ¿ 1. When m = 1, we have the
following result whose proof is similar to the proof of Lemma 1.2 in [4].

278

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

Theorem 5.5. Suppose that
h(p + q) ¡ k =k :
If tn ; yn−m ; yn−m+1 ; : : : ; yn+k−1 are given, then yn+k is uniquely determined by Eqs. (3.3) with (3.4).
6. Equations with several delays
Consider the following equation with several delays:
y′ (t) = f(t; y(t); y(t − 1 ); y(t − 2 ); : : : ; y(t − r ));
y(t) = 1 (t);

t 6 0:

t ¿ 0;
(6.1)

Because 1 ; 2 ; : : : ; r are positive constants, there are no additional diculties with respect to (6.1).
We can similarly dene the concepts of stability and convergence in this case. All results given in
this paper can be modied easily to this more general situation. But we do not list them here for
the sake of brevity.
Acknowledgements
The authors would like to thank an anonymous referee for his many valuable suggestions and
carefully correcting the preliminary version of the manuscript.
References
[1] V.K. Barwell, Special stability problems for functional equations, BIT 15 (1975) 130–135.
[2] A. Bellen, M. Zennaro, Strong contractivity properties of numerical methods for ordinary and delay dierential
equations, Appl. Numer. Math. 9 (1992) 321–346.
[3] J.C. Butcher, The Numerical Analysis of Ordinary Dierential Equations, Wiley, Chichester, 1987.
[4] G. Dahlquist, Error analysis for a class of methods for sti nonlinear initial value problems, Num. Anal. Dundee,
1975, Lecture Notes in Mathematics, vol. 506, Springer, Berlin, 1976, pp. 60 –74.
[5] G. Dahlquist, G-stability is equivalent to A-stability, BIT 18 (1978) 384 – 401.
[6] K. Dekker, J.G. Verwer, Stability of Runge–Kutta Methods for Sti Nonlinear Dierential Equations, CWI
Monographs 2, North-Holland, Amsterdam, 1984.
[7] R. Frank, J. Schneid, C.W. Ueberhuber, The concept of B-convergence, SIAM J. Numer. Anal. 18 (1981) 753–780.
[8] J. Hale, Theory of Functional Dierential Equations, Springer, New York, 1977.
[9] E. Hairer, G. Wanner, Solving Ordinary Dierential equations II, Springer, Berlin, 1991.
[10] C.M. Huang, A-stability is equivalent to B-convergence, Chinese J. Eng. Math. 12 (1995) 119–122.
[11] W.H. Hundsdorfer, B.I. Steininger, Convergence of linear multistep and one-leg methods for sti nonlinear initial
value problems, BIT 31 (1991) 124 –143.
[12] K.J. in’t Hout, A new interpolation procedure for adapting Runge–Kutta methods to delay dierential equations, BIT
32 (1992) 634 – 649.
[13] K.J. in’t Hout, Stability analysis of Runge–Kutta methods for systems of delay dierential equations, IMA J. Numer.
Anal. 17 (1997) 17–27.
[14] K.J. in’t Hout, M.N. Spijker, Stability analysis of numerical methods for delay dierential equations, Numer. Math.
59 (1991) 807–814.

C.M. Huang et al. / Journal of Computational and Applied Mathematics 103 (1999) 263–279

279

[15] S.F. Li, B-convergence of general linear methods, Proc. BAIL-V Inter. Congress, Shanghai, 1988, pp. 203–208.
[16] S.F. Li, Stability and B-convergence of general linear methods, J. Comput. Appl. Math. 28 (1989) 281–296.
[17] S.F. Li, Stability criteria for one-leg methods and linear multistep methods, Natur. Sci. J. Xiangtan Univ. 9 (1987)
21–27.
[18] S.F. Li, Theory of Computaional Methods for Sti Dierential Equations, Human Science and Technology Publisher,
Changsha, 1997.
[19] M.Z. Liu, M.N. Spijker, The stability of the -methods in the numerical solution of delay dierential equations, IMA
J. Numer. Anal. 10 (1990) 31– 48.
[20] L. Torelli, Stability of numerical methods for delay dierential equations, J. Comput. Appl. Math. 25 (1989) 15–26.
[21] D.S. Watanabe, M.G. Roth, The stability of dierence formulas for delay dierential equations, SIAM J. Numer.
Anal. 22 (1985) 132–145.
[22] M. Zennaro, P-stability of Runge–Kutta methods for delay dierential equations, Numer. Math. 49 (1986) 305–318.
[23] M. Zennaro, Asymptotic stability analysis of Runge–Kutta methods for nonlinear systems of delay dierential
equations, Numer. Math. 77 (1997) 549–563.
[24] C. Zhang, S. Zhou, Nonlinear stability and D-convergence of Runge–Kutta methods for DDEs, J. Comput. Appl.
Math. 85 (1997) 225–237.