DSpace at VNU: Improved parallel-iterated pseudo two-step RK methods for nonstiff IVPs

DSpace at VNU: Improved parallel-iterated pseudo two-step RK methods for nonstiff IVPs tài liệu, giáo án, bài giảng , lu...

Improved parallel-iterated pseudo two-step

Nguyen Huu Conga,∗, Le Ngoc Xuanb

aSchool of Graduate Studies, Vietnam National University, Hanoi G7, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam

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

Available online 16 January 2007

Abstract

The aim of this paper is to consider a parallel predictor–corrector (PC) iteration scheme for a general class of pseudo two-step Runge–Kutta methods (PTRK methods) of arbitrary high-order for solving first-order nonstiff initial-value problems (IVPs) on

parallel computers Starting with an s-stage pseudo two-step RK method of order p∗with w implicit stages, we apply a highly

parallel PC iteration process in PE(CE) m Emode The resulting parallel PC method can be viewed as a parallel-iterated pseudo two-step Runge–Kutta method (PIPTRK method) with an improved (new) predictor formula and therefore will be called the improved

PIPTRK method (IPIPTRK method) The IPIPTRK method uses an optimal number of processors equal to w  p∗/2 Numerical experiments show that the IPIPTRK methods proposed in this paper are superior to the efficient sequential DOPRI5 and DOP853 codes and parallel PIRK methods available in the literature

©2006 IMACS Published by Elsevier B.V All rights reserved

MSC: 65M12; 65M20

Keywords: RK methods; PC methods; Parallelism

1 Introduction

The arrival of parallel computers influences the development of numerical methods for the solution of the nonstiff initial value problem (IVP) for systems of first-order ordinary differential equations (ODEs)

y(t )= ft, y(t )

where y, f∈ Rd The most efficient numerical methods for solving this problem 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., [16,18,19] In order to exploit the facilities of parallel computers, a number of parallel explicit methods have been investigated in e.g., [1–4,6–9,11–15,20–22] A common challenge in the latter mentioned works is to reduce, for a given order of

accuracy, the required number of effective sequential f-evaluations per step, using parallel processors In the present

paper, we propose a class of parallel predictor–corrector (PC) iteration methods based on the class of pseudo two-step

✩ This work was supported by NRPFS.

* Corresponding author.

E-mail address: congnh@vnu.edu.vn (N.H Cong).

0168-9274/$30.00 © 2006 IMACS Published by Elsevier B.V All rights reserved.

doi:10.1016/j.apnum.2006.11.005

RK methods (PTRK methods) recently proposed in [10] An s-stage PTRK method using w implicit stages, v = s − w explicit stages is of step point and stage order both at least equal to s with any integer pair w, v, with w + v = s It

is always zero-stable and can attain the step point order p∗= s + 1 (see Section 2) Applying a highly parallel PC

iteration scheme to the PTRK methods gives us the parallel PC methods that are similar to the parallel-iterated pseudo two-step RK methods (PIPTRK methods) proposed in [12] with an improved (new) predictor formula Therefore, the

resulting PC iteration methods will be termed improved parallel-iterated PTRK methods (IPIPTRK methods) (see

Section 3)

Although, for a given number of processors, the order of the IPIPTRK methods used for the numerical experiments

in this paper is equal to that of the PIRK methods, their rate of convergence is better, their predictor formula is more accurate, so that their efficiency is expected to be increased (see Section 4) The increased efficiency is demonstrated

in Sections 4.1 and 4.2 by comparing numerical results of the IPIPTRK methods with those of PIRK and sequential explicit RK methods available in the literature

In the following sections, for the sake of simplicity of notation, we assume that the IVP (1.1) is a scalar problem However, all considerations below can be straightforwardly extended to a system of ODEs

2 PTRK corrector methods

The PTRK corrector methods were firstly considered in [10] In this section, we give an overview on these PTRK

methods Let collocation vector c be partitioned into two subvectors cvand cw that is c= (c T

v , c T w ) T, then a general

s-stage PTRK method based on c for a scalar problem is defined by

Vn = u nev + hA vv f (t n−1ev + hc v , V n−1) + hA vw f (t n−1ew + hc w , W n−1), (2.1a)

Wn = u new + hA wv f (t nev + hc v , V n ) + hA ww f (t new + hc w , W n ), (2.1b)

u n+1= u n + hb T

v f (t nev + hc v , V n ) + hb T

where, u n+1≈ y(t n+1) , A ij are i × j method parameter matrices, b j, dj and cj are j -dimensional method

pa-rameter vectors, ej is the j -dimensional vector with unit entries, (for i, j = v, w, v + w = s) V n is called the

explicit stage subvector representing the numerical approximation to the exact solution vector y(t nev+ cvh) =

[y(t n + c1h), , y(t n + c v h)]T and Wn is called the implicit stage subvector representing the numerical

approxi-mation to the exact solution vector y(t new+ cw h) = [y(t n + c v+1h), , y(t n + c s h)]T Furthermore, in (2.1) and

elsewhere in this paper, we use for any two vectors ξ = (ξ1, , ξ s ) T , η = (η1, , η s ) T and any scalar function f the

notation f (ξ , η) := [f (ξ1, η1), , f (ξ s , η s )]T

The method parameter matrices A ij and vectors bj (for i, j = v, w, v + w = s), will be determined by order

conditions (see Theorem 2.1 below) This PTRK method is conveniently specified by the tableau

A vv A vw cv O vv O vw

O wv O ww cw A wv A ww

u n+1 bT v bT w

(2.2)

In order to start the PTRK method (2.1), an appropriate starting procedure is needed for generating sufficiently

accu-rate starting stage vectors V0, W0and step point value u1from u0= y0 This can be done, for example, by using an appropriate PIRK method considered in [21] or a sequential RK code with dense output

The s-stage PTRK method (2.1) consists of v explicit stages and w implicit stages Its order can be studied in the same way as the order of RK methods Thus suppose that u n = y(t n ), Vn−1= y(t n−1ev + hc v ) and Wn−1=

y(tn−1ew + hc w ), then we have the following order definition (see [10]):

Definition 2.1 The PTRK method (2.1) is said to be of order p∗if

y(tn+1) − u n+1= Oh p∗+1

, and stage order q∗= min{p∗, q1, q2} if in addition,

y(t nev + hc v )− Vn = Oh q1 +1

, y(t e + hc )− W = Oh q2+1

.

The following theorem gives the order conditions for PTRK methods (see [10, Theorem 2.1]).

Theorem 2.1 If the function f is Lipschitz continuous, and if

(Avv, Avw)(c − e) j−1=c

j v

(Awv , Aww )c j−1=c

j w



bT v , b T w

cj−1=1

then the PTRK method (2.1) has order p∗= min{p, q1+ 1, q2+ 1} and stage order q∗= min{p∗, q1, q2} for any

collocation vector c with distinct abscissas and for any integer pair v, w with w + v = s.

In order to express the parameter matrices A vv , A vw , A wv , A ww and vectors bv, bw explicitly in terms of the

collocation vector c= (c T , c T w ) T, we define the following matrices and vectors

P v:=



cv

1,

c2

2,

c3

3,

c4

4, ,

cs v

s



, P w:=



cw

1 ,

c2w

2 ,

c3w

3 ,

c4w

4 , ,

cs w

s



,

R:=e, c, c2, c3, , c s−1

, gT :=

 1

1,

1

2, ,

1

s



,

Q:=e, (c − e), (c − e)2, , (c − e) s−1

Then the order conditions (2.3) in Theorem 2.1 for q1= q2= p = s can be presented in the form (cf [10])

(Avv, Avw)Q = P v, (Awv, Aww)R = P w , 

bT v , b T w

From (2.4) the parameter matrices and vectors of the PTRK method (2.1) can be expressed as follows

(Avv, Avw) = P vQ−1, (Awv , Aww) = P wR−1, 

bT v , b T w

In view of Theorem 2.1, it follows from (2.5) that

y(tnev + hc v)− Vn = Oh s+1

, y(tnew + hc w )− Wn = Oh s+1

, y(tn + h) − u n+1= Oh p+1

.

(2.6)

By virtue of Theorem 2.1, p = s for any collocation vector c with distinct abscissas With a special choice of the vector c, it is possible to increase the order p beyond s (superconvergence) by satisfying an orthogonality relation

(see e.g., [19, p 212]) as stated in the following theorem:

Theorem 2.2 An s-stage PTRK method defined by (2.1) is of step point order p∗= s and of stage order q∗= s if the parameter matrices Avv , A vw , A wv , Aww and vectors bv, bw, of the method satisfy the relations (2.5) (equivalently (2.4)) for any collocation vector c with distinct abscissas and for any integer pair v, w with w + v = s It has step point order p∗= s + 1 if in addition

Pj ( 1) = 0, P j (x):=

x



0

ξ j−1s

i=1

(ξ − c i ) dξ, j = 1, , k, holds for k  1.

The proof of this theorem can be found in [10] Theorem 2.2 indicates that an s-stage PTRK method, can attain the step point order p∗= s + 1 According to the analysis of the local errors in this section, the starting vectors V0, W0

and value u should be of order s + 1 and p∗+ 1 that is

y(t0ev + hc v)− V0= Oh s+1

, y(t0ew + hc w )− W0= Oh s+1

, y(t1) − u1= Oh p∗+1

.

Since the PTRK method (2.1) is of two-step nature, the property of zero-stability is an important requirement One can ask whether PTRK methods are zero-stable The following theorem gives the answer to this question

Theorem 2.3 The PTRK methods based on any collocation vectors c with distinct abscissas are always zero-stable

for any integer pair v, w with v + w = s.

This theorem was proved in [10] by writing the PTRK methods in the one-step form of a general linear method (see e.g., [5])

3 IPIPTRK methods

In this section, by applying a parallel PC iteration scheme to the PTRK methods defined by (2.1), we arrive at the

following PC iteration method (in PE(CE) m Emode):

Vn = y nev + hA vv f (t n−1ev + hc v , V n−1) + hA vw f

t n−1ew + hc w , W (m) n−1



W( n 0) = y new + hB wv f (t nev + hc v , V n ) + hB ww f

t n−1ew + hc w , W (m) n−1



W(j ) n = y new + hA wvf (tnev + hc v, Vn) + hA wwf

tnew + hc w, W (j −1)

n



, j = 1, , m, (3.1c)

y n+1= y n + hb T

v f (t nev + hc v , V n ) + hb T

w f

t new + hc w , W (m) n 

where the matrices B vw and B wwin (3.1b) are determined by order conditions given in Section 3.1 This PC iteration method (3.1) is similar to a PIPTRK method considered in [12] and becomes the original PIPTRK method if the

“improved” predictor formula (3.1b) in (3.1) is replaced with the original one used in [12]:

W( n 0) = y new + hB wv f (t n−1ev + hc v , V n−1) + hB ww f

t n−1ew + hc w , W (m) n−1



.

The PC method (3.1) therefore, will be termed the improved PIPTRK method (IPIPTRK method)

As for every explicit method, the computational cost of the method (3.1) is measured by the number of

se-quential f-evaluations per step Notice that v components of f (t nev + hc v , V n ) and w components of f (t new+

hcw , W j−1

n ), j = 1, , m + 1, can be computed in parallel, provided that max(v, w) processors are available Since

f (tn−1ev + hc v, Vn−1) and f (t new + hc w, Wn−1) can be reused, in general, we need m+ 2 sequential f-evaluations

per step

3.1 Order conditions for the predictor

The sth order conditions for the predictor formula (3.1b) can be derived by replacing V n , y n, Wn−1 and W( n 0)

by the exact solution values y(t nev + hc v ) , y(t n ) , y(t n−1ew + hc w ) = y(t new + h(c w− ew )) , and y(t new + hc w ), respectively On substitution of these exact solution values into (3.1b), we are led to

y(t new + hc w ) − y(t n )e w − hB wv y(t

nev + hc v ) − hB ww y

t new + h(c w− ew )

= Oh s+1

Using a Taylor expansion for the sufficiently smooth function y(t) in the neighborhood of t n, we can expand the

left-hand side of (3.2) in powers of h and obtain the order conditions (cf., e.g., [8,10,12])

cj w − jB wvcj−1

v + B ww (c w− ew ) j−1

or equivalently

B wvcj−1

v + B ww (c w− ew ) j−1=c

j w

The conditions (3.3) determine the matrix (B wv , Bww) By using the matrices P w defined in Section 2 and the new matrix

Q1=



v

ew (c w− ew ) (c w− ew )2 (c w− ew ) s−1



,

these conditions (3.3) can be written in the form

Suppose that the following condition is satisfied:

Since the condition (3.5a) implies that the matrix Q1is nonsingular, the relation (3.4) leads us to the explicit expression

of the matrix (B wv , B ww )as follows

(B wv , B ww ) = P w Q−1

If (3.4) (equivalently (3.3)) is satisfied, then we have

Wn− W( 0)

n =Wn − y(t new + hc w )

+y(tnew + hc w )− W( 0)

n

= Oh s+1

Since the function f is Lipschitz continuous and each iteration in (3.1) raises the order of the iteration error by 1,

using (3.6) we can obtain the following order relations:

Wn− W(m)

n = Oh m +s+1

,

u n+1− y n+1= hb T

w



f (W n ) − fW(m) n 

= Oh m +s+2

, y(t n+1) − y n+1=y(t n+1) − u n+1

+ [u n+1− y n+1] = Oh p∗+1

+ Oh m +s+2

, where, p∗is the order of the PTRK corrector method (2.1) Since condition (3.5) implies the relations (3.3)

(equiva-lently (3.4)), we have the following theorem:

Theorem 3.1 If the PTRK corrector method (2.1) has step point order p∗, and if the conditions (3.5) are satisfied,

then the IPIPTRK method (3.1) has step point order p∗∗= min{p∗, m + s + 1}, for any collocation vector c with

distinct abscissas.

3.2 Rate of convergence

The rate of convergence of the IPIPTRK method (3.1) is defined by using the model test equation y(t ) = λy(t),

where λ runs through the eigenvalues of the Jacobian matrix ∂f/∂y (cf., e.g., [7,11,14,20]) Applying the IPIPTRK

method (3.1) to this model test equation, we obtain the iteration error equation

W(j ) n − Wn = zA ww



W(j −1)

Hence, with respect to the model test equation, the rate of convergence is determined by the spectral radius ρ(zA ww )

of the iteration matrix zA ww Requiring that ρ(zA ww) <1, leads us to the convergence condition

|z| < 1

We shall call ρ(A ww ) the convergence factor and 1/ρ(A ww ) the convergence boundary of the IPIPTRK methods.

The freedom in the choice of the collocation vector c of PTRK corrector methods can be used for minimizing the

convergence factor ρ(A ww ) , or equivalently, for maximizing the convergence region Sconvdefined by

Sconv:= z: |z| < 1/ρ(A ww )

Specification of convergence factors and convergence boundaries for a specified class of IPIPTRK methods used in our numerical experiments is reported in Section 4

3.3 Stability regions

The linear stability of the IPIPTRK methods (3.1) is investigated by again using the model test equation y(t )=

λy(t ) , where λ∈ C−:= {z: z ∈ C, Re(z)  0} Denoting z := λh and applying (3.1) to the model test equation yields

W(m) n =I + zA ww + · · · + (zA ww ) m−1

(e w y n + zA wvVn ) + (zA ww ) mW( n 0)

=I + zA ww + · · · + (zA ww) m−1

(ewyn + zA wvVn) + (zA ww) m

ewyn + zB wvVn + zB wwW(m) n−1



=I + zA ww + · · · + (zA ww) m−1 

ew yn + zA wv



evyn + zA vvVn−1+ zA vwW(m) n−1



+ (zA ww ) m

ew y n + zB wv



ev y n + zA vvVn−1+ zA vwW(m) n−1



+ zB wwW(m) n−1

= z2

I + zA ww + · · · + (zA ww ) m−1

AwvAvv + z m+2(Aww) m BwvAvv

Vn−1

+ z2

I + zA ww + · · · + (zA ww ) m−1

A wv A vw + z m+1(A

ww ) m

zB wv A vw + B ww

W(m) n−1

+ I + zA ww + · · · + (zA ww ) m−1

(e w + zA wvev ) + z m (A ww ) m[ew + zB wvev]y n

= M (m)

21 (z)V n−1+ M (m)

22 (z)W (m) n−1+ M (m)

y n+1= y n + zb T

vVn + zb T

wW(m) n

= y n + zb T

v



zA vvVn−1+ zA vwW(m) n−1+ ev y n

+ zb T w



M21(m) (z)V n−1+ M (m)

22 (z)W (m) n−1+ M (m)

23 (z)y n

= z2bT v A vv + zb T

w M21(m) (z)

Vn−1+ z2bT v A vw + zb T

w M22(m) (z)

W(m) n−1+ 1+ zb T

vev + zb T

w M23(m) (z)

y n

= M (m)

31 (z)Vn−1+ M (m)

32 (z)W (m) n−1+ M (m)

From (3.10) we are led to the recursion

⎛

⎝

Vn

W(m) n

y n+1

⎞

⎠ = M m (z)

⎛

⎜

⎝

Vn−1

W(m) n−1

y n

⎞

⎟

where M m (z) is the (s + 1) × (s + 1) matrix defined by

Mm(z)=

⎛

⎜

zAvv zAvw ev

M21(m) (z) M22(m) (z) M23(m) (z)

M31(m) (z) M32(m) (z) M33(m) (z)

⎞

⎟

The explicit formulas of M ij (m) (z) with i = 2, 3, j = 1, 2, 3 in (3.11b) are clear from (3.10) The matrix M m (z)in

(3.11) which determines the stability of the IPIPTRK methods, will be called the amplification matrix, its spectral radius ρ(M m(z)) the stability function For a given number of iterations m, the stability region of the IPIPTRK

methods are defined as

Sstab(m):= z : ρ

Mm(z)

< 1, Re(z) 0 The real and imaginary boundaries for a given m, βre(m) and βim(m), respectively, can be defined in the familiar way

The stability pairs (βre(m), βim(m))for the IPIPTRK methods used in our numerical experiments can be found in Section 4

4 Numerical experiments

In this paper, we report numerical results for IPIPTRK methods with the number of implicit stages w = [s/2] and the number of explicit stages v = s − w, where [·] denotes the integer part function and s = 4, 5, 6 (see Section 3).

The PTRK corrector methods defined by (2.1) are based on collocation vectors c which satisfy the condition (3.5a).

We restrict our numerical experiments to the three following IPIPTRK methods:

• The first method denoted by IPIPTRK4 uses the PTRK corrector based on collocation vector c = (c T , c T w ) T with

cv = (c1, c2) T , c w = (6

5+ c1,65+ c2) T , where c1= (3 −√3 )/6 and c2= (3 +√3 )/6 are two components of the two-dimensional Gauss–Legendre collocation vector The resulting IPIPTRK4 method is of order p∗∗= 4 (see

Theorem 2.2 and Theorem 3.1) and uses pr = w = 2 processors.

• The second method denoted by IPIPTRK5 uses the PTRK corrector based on collocation vector c = (c T , c T w ) T

with cv = (c1, c2, c3) T , c w = (6

5+ c1,65+ c2) T , where c1= (4 −√6 )/10, c2= (4 +√6 )/10 and c3= 1 are three components of the three-dimensional RadauIIA collocation vector The resulting IPIPTRK5 method is of order

p∗∗= 5 (see Theorems 2.2 and 3.1) and uses pr = w = 2 processors.

• The third method denoted by IPIPTRK6 uses the PTRK corrector based on collocation vector c = (c T

v , c T w ) T with

cv = (c1, c2, c3) T , cw = (6

5+ c1,65+ c2,65+ c3) T , where c1= (5 −√15 )/10, c2= 1/2 and c3= (5 +√15 )/10

are three components of the three-dimensional Gauss–Legendre collocation vector The resulting IPIPTRK6

method is of order p∗∗= 6 (see Theorems 2.2 and 3.1) and uses pr = w = 3 processors.

We do not claim that the above chosen IPIPTRK methods are the best possible for a given order p∗∗ A further

study of this topic will be subject of future research The orders and number of processors pr of IPIPTRK4 and IPIPTRK6 methods are the same as used for the PIRK methods proposed in [21] with pr = p∗∗/2 except for the

IPIPTRK5 method with pr < p∗∗/2 A direct numerical computation reveals that the convergence factors of the

IPIPTRK methods as defined in Section 3.2 are slightly smaller than those of PIRK methods of the same order (see

Table 1, where * indicates that no PIRK method of order p= 5 is available) Table 1 also gives the convergence

boundaries defined by 1/ρ(A ww )(see Section 3.2) of the IPIPTRK methods Table 2 lists the real and imaginary stability boundaries of the IPIPTRK methods We note that all these IPIPTRK methods have no empty imaginary stability boundaries The results reported in Tables 1 and 2 show that for the IPIPTRK methods, the convergence

boundaries are larger than stability boundaries βre(m) and βim(m) for p = 4, 5, 6 and m = 1, , 6 Hence, the benefit

of large stability regions can be fully utilized As shown in the Table 2, the stability pairs of the IPIPTRK methods are sufficiently large for nonstiff problems

We shall compare the IPIPTRK methods with parallel and sequential explicit RK methods from the literature

In the numerical experiments, for the first step, the starting values V0, W0 and y1 of an IPIPTRK method will be

generated by the associated PIRK method based on the same collocation vector c as the underlying IPIPTRK method.

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 number of correct decimal digits) 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 Nstp Ignoring load balancing factors and communication times between processors in parallel methods, the comparison

of various methods in this section is based on the scaled accuracy NCD/Nseq The numerical experiments with small widely-used test problems taken from the literature below show a potential superiority of the new IPIPTRK methods

Table 1

Convergence factors and boundaries for various pth order parallel PC methods

Table 2

Stability pairs (βre (m), βim(m)) for various pth order IPIPTRK 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]) All the computations were carried out on a 15-digit precision

computer An actual implementation with stepsize strategy for large and expensive problems on a parallel machine is

a subject of further studies

4.1 Comparison with parallel PC methods

In order to see the efficiency of various parallel PC methods, we follow a dynamical strategy in all PC methods for determining 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 underlying corrector methods This leads us to the stopping criterion (cf., e.g., [7,11])

W(m)

n − W(m −1)

∞ TOL = Ch p∗

where C is a problem- and method-dependent parameter, and p∗ is order of the PTRK correctors We shall report

numerical results obtained by ones of the best parallel explicit RK methods available in the literature, that is the PIRK methods proposed in [21] and the three IPIPTRK methods constructed 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 parallel PC methods to the two body problem on the

integration interval[0, 20], with eccentricity ε = 3

10 (cf., e.g.,[21,23])

y

1(t ) = y3(t ), y1( 0) = 1 − ε,

y

2(t ) = y4(t ), y2( 0) = 0,

y

3(t )= −y4(t )

[y2

1(t ) + y2

2(t )]3/2 , y3( 0) = 0,

y

4(t )= −y2(t )

[y2

1(t ) + y2

2(t )]3/2 , y4( 0)=



1+ ε

The numerical results listed in Table 3 clearly show that the IPIPTRK methods are much more efficient than the PIRK methods of the same order and the same number of processors For two-processor parallel PC methods considered here, the IPIPTRK5 method is the best

4.1.2 Fehlberg problem

For the second numerical test, we apply the various pth order parallel PC methods to the often-used Fehlberg problem on the integration interval [0, 5] (cf.,e.g., [7,21,23])

y

1(t ) = 2ty1(t )log

max y2(t ),10−3

, y1( 0) = 1,

y

2(t ) = −2ty2(t )log

max y1(t ),10−3

with the exact solution y1(t ) = exp(sin(t2)) , y2(t ) = exp(cos(t2)) The numerical results are reported in Table 4 These numerical results show that the IPIPTRK methods are again by far superior to the PIRK methods of the same order and the same number of processors The IPIPTRK5 method is again the best of two-processor parallel PC methods

Table 3

Values of NCD/Nseq for problem (4.2) obtained by various pth order parallel PC methods with pr processors

PC methods p pr Nstp = 100 Nstp = 200 Nstp = 400 Nstp = 800 Nstp = 1600

Table 4

Values of NCD/Nseqfor problem (4.3) obtained by various pth order parallel PC methods with pr processors

PC methods p pr Nstp = 100 Nstp = 200 Nstp = 400 Nstp = 800 Nstp = 1600

Table 5

Values of NCD/Nseq for problem (4.4) obtained by various pth order parallel PC methods with pr processors

PC methods p pr Nstp = 100 Nstp = 200 Nstp = 400 Nstp = 800 Nstp = 1600

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, 20] (cf., e.g., [19, Problem JACB, p 240], also [23])

y

1(t ) = y2(t )y3(t ), y1( 0) = 0,

y

2(t ) = −y1(t )y3(t ), y2( 0) = 1,

y

The exact solution is given by the Jacobi elliptic functions y1(t ) = sn(t; k), y2(t ) = cn(t; k), y3(t ) = dn(t; k)

(see [17]) The numerical results for this problem are given in Table 5 and give rise to nearly the same conclusions as formulated in the two previous examples

4.2 Comparison with sequential methods

In Section 4.1, the IPIPTRK methods were compared with PIRK methods In this section, we shall compare these IPIPTRK 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 experiments to the comparison

of our 5th and 6th order IPIPTRK5 and IPIPTRK6 methods with two sequential codes DOPRI5 and DOP853 of 5th and 8th order, respectively, 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 [19]) They are based on the pair 5(4) and the “triple” 8(5)(3), respectively DOP853 is the new version of DOPRI8 with a “stretched” error estimator (see [19, 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 [9] and added the results in the low accuracy range obtained by IPIPTRK5 and IPIPTRK6 methods In spite of the fact that the results of the sequential codes are obtained using a stepsize strategy, whereas IPIPTRK5 and IPIPTRK6 methods are applied with fixed stepsizes, it is the IPIPTRK5 and IPIPTRK6 methods are more efficient when compared to the sequential codes

of comparable order (see Table 6) From Table 6, we can see that the IPIPTRK5 method is about two and half times cheaper than the code DOPRI5, and the IPIPTRK6 method is about two times cheaper than the code DOP853

Table 6

Comparison with sequential methods for problem (4.3)

5 Concluding remarks

In this paper, we proposed and investigated a new class of parallel PC iteration methods called improved parallel-iterated pseudo two-step RK methods (IPIPTRK methods) that are shown to be promising parallel integration methods Three numerical experiments clearly demonstrate the superiority of the IPIPTRK methods over the efficient sequential DOPRI5and DOP853 codes and parallel PIRK methods available in the literature In forthcoming papers, we will pursue the study of IPIPTRK methods via the optimal choice of the method parameters and variable stepsize control

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