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Parallel-Iterated RK-Type PC MethodsWith Continuous Output Formulas * Thiswork was partly supported by N.R.P.F.S.Nguyen Huu Cong a & Le Ngoc Xuan aa Faculty of Mathematics, Mechanics and Informatics, HanoiUniversity of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi,VietnamVersion of record first published: 15 Sep 2010.

To cite this article: Nguyen Huu Cong & Le Ngoc Xuan (2003): Parallel-Iterated RK-Type PC MethodsWith Continuous Output Formulas * This work was partly supported by N.R.P.F.S., InternationalJournal of Computer Mathematics, 80:8, 1025-1035

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Intern. J. Computer Math.,

Vol. 80, No. 8, August 2003, pp. 1025–1035

PARALLEL-ITERATED RK-TYPE PC METHODS WITH

CONTINUOUS OUTPUT FORMULAS*

NGUYEN HUU CONGy and LE NGOC XUAN

Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science,334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

(Received September 2002; Accepted March 2003)

This paper investigates parallel predictor–corrector iteration schemes (PC iteration schemes) based on collocationRunge–Kutta corrector methods (RK corrector methods) with continuous output formulas for solving nonstiffinitial-value problems (IVPs) for systems of first-order differential equations. The resulting parallel-iterated RK-type PC methods are also provided with continuous output formulas. The continuous numerical approximationsare used for predicting the stage values in the PC iteration processes. In this way, we obtain parallel PC methodswith continuous output formulas and high-accurate predictions. Applications of the resulting parallel PC methodsto a few widely-used test problems reveal that these new parallel PC methods are much more efficient whencompared with the parallel and sequential explicit RK methods from the literature.

Keywords: Runge–Kutta methods; Predictor–corrector methods; Stability; Parallelism

C.R. Categories: G.1.7

1 INTRODUCTION

The arrival of parallel computers influences the development of numerical methods for the

numerical solution of nonstiff initial-value problems (IVPs) for the systems of first-order,

ordinary differential equations (ODEs)

y0(t) ¼ f(t, y(t)), y(t0) ¼ y0, t0 � t � T , (1:1)

where y, f 2 Rd . Among various numerical methods proposed so far, the most efficient

methods for solving these problems are the explicit Runge–Kutta methods (RK methods).

In the literature, sequential explicit RK methods up to order 10 can be found in e.g.,

[13, 15, 16]. In order to exploit the facilities of parallel computers, several class of parallel

predictor–corrector methods (PC methods) based on RK corrector methods have been

investigated in e.g., [1–4, 6–8, 9, 11, 12, 17–19]. A common challenge in the latter-

mentioned papers is to reduce, for a given order of accuracy, the required number of sequen-

* This work was partly supported by N.R.P.F.S.y Corresponding author.

ISSN 0020-7160 print; ISSN 1029-0265 online # 2003 Taylor & Francis LtdDOI: 10.1080=0020716031000103321

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tial f-evaluations per step, using parallel processors. In the present paper, we investigate

a particular class of parallel-iterated RK-type PC methods based on collocation RK corrector

methods with continuous output formulas. The continuous numerical approximations can

also be used as starting stage values in the PC iteration process. In this way we obtain parallel

PC methods which will be termed parallel-iterated RK-type PC methods with continuous

output formulas (PIRKC methods). Thus, we have achieved the PC methods with dense out-

put formulas and high-order predictors. As a consequence, the resulting new PIRKC methods

require few numbers of sequential f-evaluations per step in the PC iteration process.

In Section 2, we shall consider RK corrector methods with continuous output formulas

(continuous RK corrector methods). Section 3 formulates and investigates the PIRKC

methods, where the order of accuracy, the rate of convergence and the stability property

are considered. Furthermore, in Section 4, we present numerical comparisons of PIRKC

methods with traditional parallel-iterated RK methods (PIRK methods) and sequential

explicit RK codes DOPRI5 and DOP853.

2 CONTINUOUS RK CORRECTOR METHODS

A numerical method is inefficient, if the number of output points becomes very large (cf. [16,

p. 188]). In the literature almost efficient embedded RK pairs have been provided with a

continuous output formula. For constructing PIRKC methods with such a continuous output

formula in Section 3, in this section, we consider a continuous extension of implicit RK

methods. Our starting point is an s-stage collocation implicit RK method

Yn,i ¼ un þ hXsj¼1

aijf(tn þ cjh, Yn, j), i ¼ 1, . . . , s, (2:1a)

unþ1 ¼ un þ hXsj¼1

bjf (tn þ cjh, Yn, j): (2:1b)

Let us consider a continuous output formula defined by

unþx ¼ un þ hXsj¼1

bj(x)f(tn þ cjh, Yn, j): (2:1c)

Here in (2.1), 0 � x � 2, unþx � y(tnþx), with tnþx ¼ tn þ xh and h is the stepsize. The

vector Yn ¼ (Yn,1, . . . , Yn,s)T denotes the stage vector representing numerical approxima-

tions to the exact solution vector (y(tn þ c1h), . . . , y(tn þ csh))T at nth step. The s� s matrix

A ¼ (aij), s-dimensional vectors b ¼ (bj), b(x) ¼ (bj(x)) and c ¼ (cj) are the method para-

meters matrix and vectors. By the collocation principle, the implicit RK method (2.1) is

of step point order p and stage order q both at least equal s. This method will be referred

to as the continuous RK corrector method and can be conveniently presented by the

Butcher tableau (see e.g., [5])

c A

ynþ1 bT

ynþx bT(x)

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The matrix A and the vector b are defined by the simplifying conditions C(s) and B(s),

respectively (see e.g., [5, 16]). They can be explicitly expressed in terms of the collocation

vector c as (see also [8])

A ¼ PR�1, bT ¼ gTR�1, (2:2)

where

P ¼ ( pij) ¼cji

j

!, R ¼ (rij) ¼ (c

j�1i ), g ¼ (gi) ¼

1

i

� �, i, j ¼ 1, . . . , s:

The vector b(x) in the continuous output formula (2.1c) is a vector function of x. It satisfies

the continuity conditions b(0) ¼ 0 and b(1) ¼ b and will be determined by order conditions.

For the fixed stepsize h, these order conditions can be derived by replacing unþx, un and Yn, j

in (2.1c) with the exact solution values and by requiring that the residue is of O(hsþ1). Using

Taylor expansions for sufficiently smooth function y(t) in the neighbourhood of tn, we obtain

the order conditions for determining b(x) (cf. e.g., [8])

D( j) ¼x j

j� bT(x)c j�1

� �¼ 0, j ¼ 1, . . . , s: (2:3a)

The conditions (2.3a) can be seen to be of the form

bT(x)R� gTdiag{x, x2, . . . , xs} ¼ 0T: (2:3b)

From (2.3b) the explicit expression of the vector b(x) then comes out

bT ¼ gTdiag{x, x2, . . . , xs}R�1: (2:3c)

In view of (2.2) and (2.3c), it follows that the continuity conditions for the vector b(x) are

clearly verified. It is also evident that if conditions (2.3) are satisfied, then we have the

local order relation

y(tnþx) � unþx ¼ O(hsþ1): (2:4)

This local order relation means that the continuous output formula defined by (2.1c) is of

order p� ¼ s. However, it turns out that if the step point order p of the continuous corrector

method (2.1) (order of the discrete implicit RK method defined by (2.1a) and (2.1b)) is

greater than s, it is possible to obtain the order p� ¼ sþ 1 for (2.1c), i.e., the order relation

(2.4) is global. Indeed, let us consider the global error estimate (without the local assumption

un ¼ y(tn))

y(tnþx) � unþx ¼ y(tnþx) � un � hXsj¼1

bj(x)f(tn þ cjh, Yn, j)

¼ y(tnþx) � y(tn) � hXsj¼1

bj(x)f (tn þ cjh, y(tn þ cjh)) þ [y(tn) � un]

þ hXsj¼1

bj(x)[f(tn þ cjh, y(tn þ cjh)) � f(tn þ cjh, Yn, j)]: (2:5)

PARALLEL-ITERATED RK-TYPE PC METHODS WITH CONTINUOUS OUTPUT FORMULAS 1027

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If the function f is Lipschitz continuous, then the following global order estimates hold:

y(tnþx) � y(tn) � hXsj¼1

bj(x)f(tn þ cjh, y(tn þ cjh)Þ ¼ O(hsþ1),

y(tn) � un ¼ O(hp),

hXsj¼1

bj(x)[f(tn þ cjh, y(tn þ cjh)) � f (tn þ cjh, Yn, j)] ¼ O(hsþ1):

(2:6)

Hence, p� ¼ sþ 1; if p > s. Thus, in general, we have

THEOREM 2.1 If the function f is Lipschitz continuous and if the continuous RK corrector

method (2:1) is of step point order p, then the continuous output formula defined by (2:1c)

gives rise to a continuous approximation of order p� ¼ min{ p, sþ 1}.

3 PIRKC METHODS

In this section, we consider the parallel PC iteration scheme using the continuous RK correc-

tor methods. This iteration scheme is given by

Y(0)n,i ¼ yn�1 þ h

Xsj¼1

bj(1 þ ci)f(tn�1 þ cjh, Y(m)n�1, j), i ¼ 1, . . . , s, (3:1a)

Y(k)n,i ¼ yn þ h

Xsj¼1

aijf(tn þ cjh, Yðk�1Þn, j ), i ¼ 1, . . . , s, k ¼ 1, . . . , m, (3:1b)

ynþ1 ¼ yn þ hXsj¼1

bjf(tn þ cjh, Y(m)n, j ), (3:1c)

ynþx ¼ yn þ hXsj¼1

bj(x)f(tn þ cjh, Y(m)n, j ): (3:1d)

Regarding (3.1a) as predictor and (2.1) as corrector, we arrive at a predictor–corrector

method (PC method) in PE(CE)mE mode. Since the evaluation of f(tn�1 þ cjh, Y(m)n�1, j),

j ¼ 1, . . . , s is available from the preceding step, we have in fact, a PC method in

P(CE)mE mode.

In the PC method (3.1), the predictions (3.1a) are obtained by using continuous output

formula (3.1d) from previous step. If in (3.1a) we set Y(0)n,i ¼ yn, i ¼ 1, . . . , s, the PC method

ð3:1aÞ–ð3:1cÞ becomes the original parallel-iterated RK methods considered in [18].

Therefore, we call the method (3.1), a parallel-iterated RK-type PC method with continuous

output formulas (PIRKC method). Notice that the s components f (tn þ cjh, Y(k�1)n, j ),

j ¼ 1, . . . , s can be evaluated in parallel, provided that the s processors are available, so

that the number of sequential f-evaluations per step of length h in each processor equals

s� ¼ mþ 1.

THEOREM 3.1 If the function f is Lipschitz continuous and if the continuous RK corrector

method (2:1) has step point order p, then the PIRKC method (3:1) has step point order

q ¼ min{ p, mþ sþ 1} and gives rise to a continuous approximation of order q� ¼

min{ p, sþ 1}.

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The proof of this theorem is simple. Suppose that f is Lipschitz continuous and

yn ¼ un ¼ y(tn). Since Yn,i � Y(0)n,i ¼ O(hsþ1) (see (2.4)) and each iteration raises the order

of the iteration error by 1, we obtain the following order relations

Yn,i � Y(m)n,i ¼ O(hmþsþ1), i ¼ 1, . . . , s,

unþ1 � ynþ1 ¼ hXsj¼1

bj[f(tn þ cjh, Yn, j) � f(tn þ cjh, Y(m)n, j )] ¼ O(hmþsþ2):

(3:2)

Hence, for the local truncation error of the PIRKC method (3.1), we may write

y(tnþ1) � ynþ1 ¼ [y(tnþ1) � unþ1] þ [unþ1 � ynþ1]

¼ O(hpþ1) þ O(hmþsþ2): (3:3)

This order relation (3.3) gives the step point order q as stated in Theorem 3.1 for the PIRKC

method. Furthermore, for the order q� of the continuous approximations defined by (3.1d),

we may also write

y(tnþx) � ynþx ¼ [y(tnþx) � unþx] þ [unþx � ynþx]

¼ [y(tnþx) � unþx] þ [un � yn]

þ hXsj¼1

bj(x)[f (tn þ cjh, Yn, j) � f(tn þ cjh, Y(m)n, j )] (3:4)

From (3.2), (3.3) and Theorem 2.1 we have the following global order relations

y(tnþx) � unþx ¼ O(hmin{p,sþ1})

un � yn ¼ [un � y(tn)] þ [y(tn) � yn] ¼ O(hmin{p,mþsþ1})

hXsj¼1

bj(x)[f(tn þ cjh, Yn, j) � f(tn þ cjh, Y(m)n, j )] ¼ O(hmin{p,mþsþ1}):

(3:5)

The relations (3.4) and (3.5) then complete the proof of Theorem 3.1.

Remark From Theorem 3.1, we see that by setting m ¼ p� s� 1, we have a PIRKC

method of maximum step point order q ¼ p (order of the corrector method) with minimum

number of sequential f-evaluations per step s� ¼ p� s.

3.1 Rate of Convergence

The rate of convergence of PIRKC methods is defined by using the model test equation

y0(t) ¼ ly(t), where l runs through the eigenvalues of the Jacobian matrix qf=qy (cf. e.g.,

[6, 10, 12, 17]). For this equation, we obtain the iteration error equation

Y( j)n � Yn ¼ zA[Y( j�1)

n � Yn], z :¼ hl, j ¼ 1, . . . , m: (3:6)

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Hence, with respect to the model test equation, the convergence rate is determined by

the spectral radius r(zA) of the iteration matrix zA. Requiring that r(zA) < 1, leads us to the

convergence condition

jzj <1

r(A)or h <

1

r(qf=qy)r(A): (3:7)

We shall call r(A) the convergence factor and 1=r(A) the convergence boundary of the

PIRKC method. One can exploit the freedom in the choice of the collocation vector c of

RK correctors for minimizing the convergence factor r(A), or equivalently, for maximizing

the convergence region denoted by Sconv and defined as

Sconv :¼ z: z 2 C, jzj <1

r(A)

� �: (3:8)

3.2 Stability Regions

The linear stability of the PIRKC methods (3.1) is investigated by again using the model test

equation y0(t) ¼ ly(t), where l is assumed to be lying in the left half-plane. By defining

the matrix

B ¼ (b(1 þ c1), . . . ,b(1 þ cs))T,

for the model test equation, we can present the starting vector defined by (3.1a)

Y(0)n ¼ (Y

(0)n,1, . . . ,Y(0)

n,s)T in the form

Y(0)n ¼ eyn�1 þ zBY

(m)n�1,

where z :¼ hl. Applying (3.1a)–(3.1c) to the model test equation yields

Y(m)n ¼ eyn þ zAY(m�1)

n

¼ [I þ zAþ � � � þ (zA)m�1]eyn þ (zA)mY(0)n

¼ zmþ1AmBY(m)n�1 þ [I þ zAþ � � � þ (zA)m�1]eyn þ zmAmeyn�1 (3:9a)

ynþ1 ¼ yn þ zbTY(m)n

¼ zmþ2bTAmBY(m)n�1 þ {1 þ zbT[I þ zAþ � � � þ (zA)m�1]e}yn

þ zmþ1bTAmeyn�1: (3:9b)

From (3.9) we are led to the recursion

Y(m)n

ynþ1

yn

0@

1A ¼ Mm(z)

Y(m)n�1

ynyn�1

0@

1A, (3:10a)

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where Mm(z) is the (sþ 2) � (sþ 2) matrix defined by

Mm(z) ¼

zmþ1AmB [I þ zAþ � � � þ (zA)m�1]e zmAme

zmþ2bTAmB 1 þ zbT[I þ zAþ � � � þ (zA)m�1]e zmþ1bTAme

0T 1 0

0B@

1CA: (3:10b)

The matrix Mm(z) defined by (3.8) which determines the stability of the PIRKC methods, will

be called the amplification matrix, its spectral radius r(Mm(z)) the stability function. For a

given number m, the stability region noted by Sstab(m) of the PIRKC methods is defined as

Sstab(m) :¼ {z: r(Mm(z)) < 1, Re(z) � 0}:

The real and imaginary stability boundaries for a given m, bre(m) and bim(m), respectively,

can be defined in the familiar way. These stability pairs (bre(m), bim(m)) for the PIRKC

methods used in the numerical experiments can be found in Section 4.

4 NUMERICAL EXPERIMENTS

This section will report numerical results for the PIRKC methods. We confine our considera-

tions to the PIRKC methods based on symmetric collocation vector c with odd number of

collocation points investigated in Ref. [6]. The continuous s-stage RK corrector methods

(2.1) based on these symmetric collocation vectors have the orders p ¼ p� ¼ sþ 1

(cf. [6, Theorem 2.1] and Theorem 2.1 in this paper). The collocation vectors were chosen

such that the spectral radius r(A) of the RK matrix A is minimized, so that the PIRKC

methods defined by (3.1) have ‘‘optimal’’ rate of convergence (cf. [6]). Table I below lists

the stability pairs of these specified PIRKC methods of orders p ¼ 4, 6, 8, 10. We observe

that the imaginary stability boundaries of these PIRKC methods show a rather irregular

behaviour. From Table I, we can select a whole set of PIRKC methods of step point order

up to 10 requiring 2 or 3 (m ¼ 1 or m ¼ 2) f-evaluations per step with acceptable stability

for nonstiff problems (cf. Theorem 3.1).

In the following, we shall compare the PIRKC methods with parallel and sequential expli-

cit RK methods from the literature. For the PIRKC methods, in the first step, we always use

the trivial predictions given by

Y(0)0,i ¼ y0, i ¼ 1, . . . , s:

The absolute error obtained at the end point of the integration interval is presented in the

form 10�NCD (NCD may be interpreted as the average number of correct decimal digits).

TABLE I Stability Pairs (bre(m), bim(m)) for Various pth-order PIRKC Methods.

PIRKC methods p¼ 4 p¼ 6 p¼ 8 p¼ 10

m¼ 1 (0.783, 0.000) (0.800, 0.643) (0.243, 0.000) (0.057, 0.000)m¼ 2 (1.499, 1.047) (1.226, 1.124) (0.558, 0.474) (0.245, 0.227)m¼ 3 (2.601, 2.020) (1.292, 1.228) (0.665, 0.670) (0.507, 0.511)m¼ 4 (2.630, 1.275) (1.484, 0.010) (0.986, 0.007) (0.887, 0.006)m¼ 5 (2.475, 2.124) (1.656, 0.040) (1.193, 0.031) (0.976, 0.026)m¼ 6 (2.516, 0.061) (2.049, 1.367) (1.512, 1.270) (1.206, 1.079)

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The computational efforts are measured by the values of Nseq denoting the total number of

sequential f-evaluations required over the total number of integration steps denoted by

Nstp. Ignoring load balancing factors and communication times between processors in parallel

methods, the comparison of various methods in this section is based on Nseq and the obtained

NCDs. The numerical experiments with small widely-used test problems taken from the lit-

erature below show a potential superiority of the new PIRKC methods over extant methods.

This superiority will be significant in a parallel machine if the test problems are large enough

and=or the f-evaluations are expensive (cf. e.g., [3]). In order to see the convergence beha-

viour of our PIRKC methods, we follow a dynamical strategy in all PC methods for deter-

mining the number of iterations in the successive steps. It seems natural to require that

the iteration error is of the same order in h as the local error of the corrector. This leads

us to the stopping criterion (cf. e.g., [6, 11])

kY(m)n � Y(m�1)

n k1 � TOL ¼ Chp, (4:1)

where C is a problem- and method-dependent parameter, p is the step point order of the cor-

rector method. All the computations were carried out on a 29-digit precision computer. An

actual implementation on a parallel machine is a subject of further studies.

4.1 Comparison with Parallel Methods

We shall report numerical results obtained by one of the best parallel explicit RK methods

available in the literature, that is the PIRK methods proposed in [18] and the PIRKC methods

considered in this paper. We selected a test set of three problems taken from the literature.

4.1.1 Two Body Problem

As a first numerical test, we apply the various pth-order PC methods to the two body problem

on the integration interval [0, 20], with eccentricity e ¼ 3=10 (cf. e.g., [18, 20])

y01(t) ¼ y3(t), y1(0) ¼ 1 � e,

y02(t) ¼ y4(t), y2(0) ¼ 0,

y03(t) ¼�y4(t)

[ y21(t) þ y2

2(t)]3=2, y3(0) ¼ 0,

y04(t) ¼�y2(t)

[ y21(t) þ y2

2(t)]3=2, y4(0) ¼

ffiffiffiffiffiffiffiffiffiffiffi1 þ e1 � e

r:

(4:2)

The numerical results listed in Table II clearly show that the PIRKC methods are much more

efficient than the PIRK methods of the same order. For this problem, all the PIRKC methods

need only about two or three iterations per step.

4.1.2 Fehlberg Problem

For the second numerical test, we apply the various pth-order PC methods to the often-used

Fehlberg problem on the integration interval [0, 5] (cf. e.g., [6, 18, 20])

y01(t) ¼ 2ty1(t) log(max{y2(t), 10�3}) y1(0) ¼ 1,

y02(t) ¼ �2ty2(t) log(max{y1(t), 10�3}) y2(0) ¼ e,(4:3)

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with the exact solution y1(t) ¼ exp(sin (t2)), y2(t) ¼ exp(cos (t2)). The numerical results are

reported in Table III. These numerical results show that the PIRKC methods are again by

far superior to the PIRK methods of the same order.

4.1.3 Jacobian Elliptic Functions Problem

The final numerical example is the Jacobian elliptic functions sn, cn, dn problem for the

equation of motion of a rigid body without external forces on the integration interval

[0, 18] (cf. e.g., [18, problem JACB, p. 240], also [20])

y01(t) ¼ y2(t)y3(t), y1(0) ¼ 0,

y02(t) ¼ �y1(t)y3(t), y2(0) ¼ 1,

y03(t) ¼ �0:51y1(t)y2(t), y3(0) ¼ 1:

(4:4)

The exact solution is given by the Jacobian elliptic functions y1(t) ¼ sn(t; k), y2(t) ¼ cn(t; k),

y3(t) ¼ dn(t; k) (see [14]). The numerical results for this problem are given in Table IV

and give rise to nearly the same conclusions as formulated in the two previous

examples.

TABLE III Values of NCD=Nseq for Problem (4.3) Obtained by Various pth-order Parallel PC Methods.

Nstp

PC methods p 100 200 400 800 1600 C

PIRK 4 2.7=392 4.0=842 5.2=1756 6.5=3650 7.7=7409 103

PIRKC 4 2.9=230 4.1=458 5.7=915 7.2=1826 8.8=3657 103

PIRK 6 5.2=601 7.0=1245 8.9=2542 10.7=5199 12.5=10,488 103

PIRKC 6 6.5=297 8.4=572 10.2=1114 12.1=2176 14.0=4241 103

PIRK 8 7.8=774 10.2=1603 12.6=3297 15.1=6674 17.5=13,468 103

PIRKC 8 9.4=363 11.8=680 14.7=1310 17.2=2497 19.5=4847 103

PIRK 10 9.9=942 12.9=1947 15.9=3973 18.9=8134 22.0=16,407 103

PIRKC 10 11.9=410 15.4=771 18.7=1458 22.4=2812 25.3=5390 103

TABLE II Values of NCD=Nseq for Problem (4.2) Obtained by Various pth-order Parallel PC Methods.

Nstp

PC methods p 100 200 400 800 1600 C

PIRK 4 3.1=441 3.7=905 4.9=1947 6.1=4000 7.3=8000 100

PIRKC 4 3.0=236 4.6=469 6.2=936 7.7=1869 8.9=3732 100

PIRK 6 5.0=643 7.2=1302 8.9=2637 10.5=5499 12.3=11,200 10�1

PIRKC 6 5.8=284 8.8=552 10.0=1061 12.2=2047 14.3=4089 10�1

PIRK 8 7.6=837 10.4=1686 12.8=3397 15.0=6845 17.3=13,827 10�2

PIRKC 8 8.2=342 10.6=649 14.4=1218 16.7=2397 19.2=4685 10�2

PIRK 10 9.3=926 12.8=1926 16.3=3927 19.2=8226 22.2=16,532 10�2

PIRKC 10 9.9=335 13.7=631 16.7=1200 20.0=2336 23.2=4585 10�2

PARALLEL-ITERATED RK-TYPE PC METHODS WITH CONTINUOUS OUTPUT FORMULAS 1033

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4.2 Comparison with Sequential Methods

In Section 4.1, the PIRKC methods were compared with PIRK methods. In this section, we

shall compare these PIRKC methods with some of the best sequential explicit RK methods

currently available.

In order to compare the methods of comparable order, we restricted the numerical experi-

ments to the comparison of our 8th order PIRKC method denoted by PIRKC8 with two

sequential codes DOPRI5 and DOP853 for the Fehlberg problem (4.3). These DOPRI5 and

DOP853 codes are embedded explicit RK methods due to Dormand and Prince and coded

by Hairer and Waner (see [16]). They are based on pair 5(4) and ‘‘triple’’ 8(5)(3), respec-

tively. DOP853 is the new version of DOPRI8 with a ‘‘stretched’’ error estimator (see [16,

p. 254]). These two codes belong to the most efficient currently existing sequential codes

for nonstiff first-order ODE problems. We took the best results obtained by DOPRI5 and

DOP853 given in [8] and added the results obtained by PIRKC8 method in the low accu-

racy-range. In spite of the fact that the results of the sequential codes are obtained using a

stepsize strategy, whereas PIRKC8 method is applied with fixed stepsizes, it is the

PIRKC8 method that is the most efficient (see Tab. V).

TABLE V Comparison with Sequential Methods for Problem (4.3).

Methods Nstp NCD Nseq

DOPRI5 (from [18]) 75 3.2 452162 5.3 974393 7.4 2360979 9.4 5876

2458 11.4 14,750

DOP853 (from (18]) 47 4.5 55270 6.2 825

107 8.0 1265164 10.2 1950261 12.2 3123

PIRKC8 (in this paper) 25 3.1 12750 5.9 195

100 9.4 363200 11.8 680400 14.7 1310

TABLE IV Values of NCD=Nseq for Problem (4.4) Obtained by Various pth-order Parallel PC Methods.

Nstp

PC methods p 100 200 400 800 1600 C

PIRK 4 2.3=300 5.1=800 6.3=1600 7.5=3200 8.9=6571 101

PIRKC 4 4.1=202 5.6=402 7.1=802 8.6=1602 10.1=3203 101

PIRK 6 5.1=486 7.8=1126 11.2=2345 12.5=4775 14.3=9600 100

PIRKC 6 6.5=204 8.5=404 10.5=804 12.6=1604 14.7=3205 100

PIRK 8 8.2=678 11.1=1470 14.0=3028 16.7=6195 19.1=12,540 10�1

PIRKC 8 8.2=238 11.2=484 13.9=963 16.6=1925 19.3=3844 10�1

PIRK 10 10.1=765 13.4=1655 16.8=3479 19.6=7095 23.2=14,968 10�1

PIRKC 10 11.7=207 13.5=422 16.5=862 19.8=1718 23.1=3439 10�1

1034 NGUYEN HUU CONG AND LE NGOC XUAN

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5 CONCLUDING REMARKS

In this paper, we considered parallel-iterated RK-type predictor–corrector methods with con-

tinuous output formulas which were denoted by PIRKC methods. By three numerical exam-

ples, we have shown that for a given order p of accuracy, the resulting PIRKC methods are by

far superior to the PIRK methods.

By comparing the 8th order PIRKC method (PIRKC8) with the most efficient sequential-

codes, we also have shown that the PIRKC methods are much more efficient.

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Oxford.[4] Burrage, K. and Suhartanto, H. (1997). Parallel iterated methods based on multistep Runge–Kutta mehods of

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