DSpace at VNU: Explicit pseudo two-step RKN methods with stepsize control

Explicit pseudo two-step RKN methods with stepsize control✩Nguyen Huu Cong Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Ha

Explicit pseudo two-step RKN methods with stepsize control✩

Nguyen Huu Cong

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

Hanoi, Viet Nam

Abstract

This paper is devoted to variable stepsize strategy implementations of a class of explicit pseudo two-step Runge– Kutta–Nyström methods of arbitrarily high order for solving nonstiff problems for systems of special second-order differential equations The constant stepsize explicit pseudo two-step Runge–Kutta–Nyström methods are developed into variable stepsize ones and equipped with embedded formulas giving a cheap error estimate for stepsize control By two examples of widely-used test problems, a pseudo two-step Runge–Kutta–Nyström method

of order 8 implemented with variable stepsize strategy is shown to be much more efficient than parallel and sequential codes available in the literature With stringent error tolerances, this new explicit pseudo two-step Runge–Kutta–Nyström method is even superior to sequential codes in a sequential computer  2001 IMACS Published by Elsevier Science B.V All rights reserved.

Keywords: Runge–Kutta–Nyström methods; Two-step Runge–Kutta–Nyström methods; Embedded formulas;

Parallelism

1 Introduction

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

a nonstiff initial value problem (IVP) for systems of special second-order ordinary differential equations (ODEs)

y

(t) = f
t, y(t)

, y(t0) = y0, y
(t0) = y

The most efficient numerical methods for solving this problem are the explicit Runge–Kutta–Nyström methods (RKN methods) In the literature, sequential explicit RKN methods up to order 11 can be found

in, e.g., [12–17,19,23] In order to exploit the facilities of parallel computers, several classes of parallel explicit methods have been investigated, for example, in [2–7,9,10] 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 our previous work [10], we have considered a

✩

This work was partly supported by DAAD, N.R.P.F.S and QG-96-02.

E-mail address: nhcong@ncst.ac.vn (N.H Cong).

0168-9274/01/$ – see front matter  2001 IMACS Published by Elsevier Science B.V All rights reserved.

PII: S 0 1 6 8 - 9 2 7 4 ( 0 1 ) 0 0 0 3 1 - 9

general class of explicit pseudo two-step RKN methods (EPTRKN methods) for solving problems of the

form (1.1) A general s-stage (constant stepsize) EPTRKN method based on an s-dimensional collocation

vector c = (c1, , c s )Twith distinct abscissas c i has the form

Y n = e ⊗ y n + hc ⊗ y

n + h2(A ⊗ I)F (t n−1e + hc, Y n−1), (1.2a)

y n+1= y n + hy

n + h2

bT⊗ IF (t n e + hc, Y n ),

y n+1= y

n + h
dT⊗ IF (t n e + hc, Y n ),

(1.2b)

where Y n = (Y n,i ) and F (t n e + hc, Y n ) = (f (t n + c i h, Y n,i )), both are sd-dimensional vectors This

method has been specified by the tableau

y n+1 bT

y
n+1 dT

.

The (constant) s-dimensional vectors b and d are the parameters of the generating implicit RKN method,

s × s matrix A is given by (see [10, Section 2.1])

A = P Q−1, P = (p ij )=



c j i+1

j+ 1



, Q = (q ij )=
j (c i − 1) j−1 

,

i = 1, , s, j = 1, , s.

(1.3)

The method (1.2) is of order p = min{p∗, s + 2} and stage order q = s, where p∗ is the order of the

generating implicit RKN method (cf [10, Theorems 2.1 and 2.2]) The number of f -evaluations per step

equals s in a sequential implementation and equals 1 in a parallel implementation using s processors.

This class of EPTRKN methods implemented with constant stepsize was shown to be very efficient for the solution of problems with stringent accuracy demand (cf [10, Section 3])

In the present work, we equip the EPTRKN methods with the ability to change the stepsize in

an implementation with stepsize control Since the EPTRKN methods are of a two-step nature, we consider the method with (variable) parameters which are functions of stepsizes (see Section 2) For

a practical error estimation used in a stepsize selection, an approach for constructing embedded formulas

is discussed in Section 3 Notice that for EPTRKN methods, embedded formulas are provided without

additional f -evaluations Finally, in Section 4, we present numerical comparisons of a variable stepsize

strategy EPTRKN method with the codes DOPRIN, ODEX2 and PIRKN currently available for two widely used test examples taken from the literature

2 Variable stepsize EPTRKN methods

It is well known that an efficient integration method must be able to change stepsizes Because EPTRKN methods are of a two-step nature, there is an additional difficulty in using these methods with variable stepsize mode There exist in principle two approaches for overcoming this difficulty (cf., e.g., [1,

p 44; 20, p 397]):

• interpolating past stage values;

• deriving methods with variable parameters.

The first approach using polynomial interpolation to reproduce the starting stage values for the new step involves with computational cost which increases as the dimension of the problem increases, while for the second approach, the computational cost is independent of the dimension of the problem For this reason, the variable parameter approach is more feasible and robust Thus, we are confined to the second approach and consider the EPTRKN method

Y n = e ⊗ y n + h n c ⊗ y
n + h2

n (A n ⊗ I)F (t n−1e + h n−1c, Y n−1), (2.1a)

y n+1= y n + h n y
n + h2

n

bT⊗ IF (t n e + h n c, Y n ),

y n+1= y

n + h n

dT⊗ IF (t n e + h n c, Y n ),

(2.1b)

with variable stepsize h n = t n+1 − t n and variable parameter matrix A n Here as in (1.2), F (t n−1e+

h n−1c, Y n−1) = (f (t n−1+ c i h n−1, Y n −1,i )) and F (t n e + h n c, Y n ) = (f (t n + c i h n , Y n,i )) The order and

stage order of a variable stepsize EPTRKN method is defined in the same way as in the case of

constant stepsize EPTRKN methods (cf [10, Definition 2.1]) The matrix A n in the method (2.1) can

be determined by order conditions as a matrix function of the stepsize ratios following [8] The (s +

1)-order conditions can be derived by replacing Y n−1, y n and Y n in (2.1a) with the exact solution values

y(t n−1e + h n−1c), y(t n ) and y(t n e + h n c), respectively, that is

y(t n e + h n c) − e ⊗ y(t n ) − h n c ⊗ y
(t

n ) − h2

n (A n ⊗ I)y

(t

n−1e + h n−1c)= O
h s n+2

Let us suppose that the stepsize ratio h n / h n−1 is bounded from above (i.e., h n / h n−1
Ω), then along

the same lines of [10, Section 2.1], using Taylor expansions, we can expand the left-hand sides of (2.2)

in powers of h n and obtain the order conditions for determining A ngiven by

C (j +1)= 1

(j + 1)!



h n

h n−1

j−1

c j+1− j (j + 1)A n (c − e) j−1 

= 0, j = 1, , s. (2.3a) The condition (2.3a) can be written in the form (cf (1.3))

A n Q − P diag



1, h n

h n−1, ,



h n

h n−1

s−1 

which gives the explicit expression of A ndefined as

A n = P diag



1, h n

h n−1, ,



h n

h n−1

s−1 

The following lemma can easily be deduced from (2.3c)

Lemma 2.1 For the variable stepsize EPTRKN method (2.1), the variable parameter matrix A n is uniformly bounded whenever the stepsize ratio h n / h n−1is bounded from above.

For h n / h n−1
Ω, the principal error vector C (s +2)is also uniformly bounded Consequently, similarly

to the order considerations for a general variable stepsize multistep method (cf., e.g., [20, p 401]), the relations (2.3) imply that locally

Y (t n e + h n c) − Y n= O
h s+2

.

Along the lines of the proof of Theorem 2.1 and Theorem 2.2 in [10], we have that if the function f is

Lipschitz continuous and if the condition of Lemma 2.1 is satisfied then at t n+1

y(t n+1) − y n+1= O
h p n+1

+ O
h s n+4

,

y
(t n+1) − y

n+1= O
h p n+1

+ O
h s n+3

, where p is the order of the associated constant stepsize EPTRKN method Hence, the order and stage

order of the variable stepsize EPTRKN method defined by (2.1) and (2.3c) is identical with those of the associated constant stepsize EPTRKN method (see [10, Theorem 2.2], also Section 1) Thus we have:

Theorem 2.1 An s-stage variable stepsize EPTRKN method defined by (2.1) with variable matrix A n

defined by (2.3c) is of order p = s and of stage order q = s for any collocation vector c with distinct

abscissas c i if h n / h n−1 is bounded from above It has stage order q = s + 1 and order p = s + 1 or

p = s + 2 if in addition the orthogonality relation

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

x

0

ξ j−1

s

i=1

(ξ − c i ) dξ ,

is satisfied for j = 1 or j  2, respectively.

Remark 2.1 The condition h n / h n−1
Ω is a reasonable assumption for a numerical code.

Remark 2.2 Zero-stability property of an EPTRKN method is independent of the method parameters

(see [10, Section 2.2]) so that the variable stepsize EPTRKN methods are always stable

3 Embedded EPTRKN methods

With the aim to have a cheap error estimate for stepsize control in an implementation of EPTRKN

methods, in parallel with the pth-order method (2.1), we consider a second pth-order EPTRKN method

based on collocation vectorc = ( c 1, , c˜s )Tof the form

Y n=e⊗y n + h nc⊗y
n + h2

n
A ⊗ I

F

t n−1 e + h n−1 c, Yn−1, 

y n+1=y n + h ny
n + h2

n
bT

⊗ IF

t ne + h nc, Yn, 

y
n+1=y
n + h n
dT

⊗ IF

t ne + h nc, Yn,

(3.1)

where, p > p, the vector c is a subvector of the vector c, i.e.,{c 1, ,c ˜s } ⊂ {c1, , c s} By introducing

new parameter vectorsb = ( b1, , bs )Tandd= ( d1, , ds )Twhich are defined according to

if c i =c j , then b i=bj , di =dj , j = 1, ,s,

else b i = 0, di = 0, i = 1, , s, (3.2)

we obtain an embedded formula without additional f -evaluations given by

y n+1= y n + h n y
n + h2

n
bT

⊗ IF (t n e + h n c, Y n ),

y
n+1= y

n + h n
dT

⊗ IF (t n e + h n c, Y n ).

(3.3)

Theorem 3.1 If the function f is Lipschitz continuous, then the numerical approximations at t n+1

defined by (1.2b) and by (3.3) locally satisfy the order relation

y n+1−yn+1= O
h n ˆp+1

,

y
n+1−y
n+1= O
h n ˆp+1

.

(3.4)

Proof As the EPTRKN method (3.1) has order p less than order p of the EPTRKN method (1.2), we

may write

y n+1−yn+1= (y n+1−y n+1) + ( yn+1−y n+1)= O
h n ˆp+1

+ ( yn+1−yn+1),

y
n+1−y
n+1=
y
n+1−y
n+1

+
y
n+1−y
n+1

= O
h n ˆp+1

+
y
n+1−y
n+1

.

(3.5a)

Since the function f is Lipschitz continuous, from the definition of the vectorsb andd in (3.2) we have 

y n+1−yn+1= ( yn − y n )+ O
h n ˜s+4

,

y
n+1−y
n+1= ( y
n − y

n )+ O
h n ˜s+3

.

(3.5b)

The relations (3.5) then prove Theorem 3.1 ✷

Thus, for a practical error estimation used in a stepsize selection we have the embedded EPTRKN method given by (2.1a), (2.1b) and (3.3) which can be specified by the tableau

y n+1 bT

y
n+1 dT

y n+1 bT

y
n+1 d T

.

The local error estimate in various vector norms is then defined by (3.4) By this approach of constructing embedded EPTRKN methods, there exist several embedded formulas for an EPTRKN method

The “lower order estimator” approach for embedded formulas described above does not give

asymptotically correct error estimates However, for this approach no additional sequential f -evaluations

are required and the resulting embedded pair can be implemented on only s processors.

Further, let us consider another approach of error estimates Since all EPTRKN methods have the same

sequential cost of one f -evaluation per step, in a parallel environment, it is possible, at no additional

sequential cost to compute asymptotically correct error estimates by using parallel processors For such

a purpose we define the embedded method of the form (3.1), wherep is sufficiently greater than p (with

s sufficiently greater than s) Then, the following order relations

y n+1−yn+1= O
h p n+1

,

y
n+1−y
n+1= O
h p n+1

give an asymptotically correct error estimate of local order p+ 1 (order of the original EPTRKN method

(2.1)) For this latter approach, the number of processors for implementing the embedded pair {(2.1),

(3.1)} is s+s Thus, with respect to the former “lower order estimator” approach,s additional processors

are needed The latter approach is also involved with constructing embedded pairs of EPTRKN methods with reasonable orders and stability properties This could be a subject of a later work

In the section of numerical experiments below, we confine our consideration to the simple “lower

order estimator” approach which is similar to that applied in other sequential and parallel codes used in the numerical comparisons and which does not require any additional processors for implementation

4 Numerical experiments

In this section we shall report the numerical results obtained by a new (parallel) EPTRKN method of orders 8, two sequential codes DOPRIN, ODEX2 and a parallel code PIRKN taken from the literature DOPRIN is a code based RKN pair 7(6) due to Dormand and Prince which can be found in the first edition of [20] ODEX2 is an extrapolation code for special second-order ODEs of the form (1.1) It uses variable order and variable stepsize and is recognized as being one of the most efficient sequential integrators for nonstiff problems of the form (1.1) (see [20, p 484]) PIRKN is a parallel code based

on the PIRKN method of order 12 taken from [25] The eighth-order EPTRKN method is based on the collocation vector

c8= (0.057, 0.277, 0.584, 0.860, 1.000, 1.277, 1.584, 1.860)T, (4.1a) with an embedded formula of order 7 based on

c6= (0.057, 0.277, 0.584, 0.860, 1.277, 1.584, 1.860)T. (4.1b) Notice that the choice of the collocation vector in (4.1a) minimizes the principal error terms for some stage approximated values (cf [10, Theorem 2.4]) and gives slightly larger stability boundary No special effort has been made to optimize the parameters of the above method An optimal choice of the method parameters was beyond the scope of this work The stability interval of the EPTRKN method defined by

(4.1a) is numerically calculated to be ( −0.596, 0) In term of considering stability of a method, it is the

scaled stability region and not the stability region that is significant (cf., e.g., [1, p 198]) The stability region of an EPTRKN method is at the same time the scaled stability region With this stability interval, the associated EPTRKN method is expected to be efficient for solving problem (1.1) especially with a stringent accuracy demand

The embedded EPTRKN pair 8(7) defined by (4.1) is implemented using local extrapolation and a starting procedure based on corrections until convergence of eight-stage direct collocation RKN method

based on c8 (cf [21]) This embedded pair 8(7) gives an estimate of local truncation error of order 8 given by

LTE= y n+1−yn+12

 2

+
y

n+1−y
n+1

2

 2

= O
h8n

where  · 2 denotes the Euclidean norm The stepsize strategy is similar to the one implemented by Sommeijer [25] in PIRKN codes which is also implemented in DOPRI5, DOP853 by Hairer and Wanner [20] A step is accepted when LTE
TOL and rejected otherwise The new stepsize h n+1 is chosen as

h n+1= h n· min2, max

0.5, 0.8 · (TOL/LTE) 1/8

The constants 2 and 0.5 serve to keep the stepsize ratios h n+1/ h n to be in the interval [0.5, 2].

This new EPTRKN method of order 8 will be denoted by EPTRKN8 Furthermore, in the tables of

numerical results, NSFCN and NPFCN denote the number of f -evaluations in sequential and parallel

implementation modes, NCD is the number of correct decimal digits, NSTEP and NREJCT are the total number of integration steps and of rejected ones, respectively Our implementation of EPTRKN8 and ODEX2 has been carried out on a number of test examples They all lead to similar observations and therefore we report here only on the results of two examples The results of DOPRIN and PIRKN codes are reproduced from [25]

4.1 Nonlinear Fehlberg problem

For the first numerical test, we apply the various codes ODEX2, DOPRIN, PIRKN and EPTRKN8 method to the well-known nonlinear Fehlberg problem (cf., e.g., [12,13,15,16])

d2y(t)

dt2 =









y2

1(t) + y2

2(t)

2

y12(t) + y2

2(t)

−4t2







y(t),

y


π/2

= (0, 1)T, y



π/2

=
−2π/2, 0 T

π/2
t
10, (4.4)

with highly oscillating exact solution given by y(t) = (cos(t2), sin(t2))T

The numerical results for this problem are listed in Table 1 We see from this table that in parallel implementation mode, the EPTRKN8 method is the most efficient With stringent error tolerances EPTRKN8 method is even superior to ODEX2 and DOPRIN in a sequential computer

4.2 Newton’s equation of motion problem

The second numerical example is the two-body gravitational problem for Newton’s equation of motion (see [24, p 245])

d2y1(t)

y12(t) + y2

2(t) 3, d

2y2(t)

y12(t) + y2

2(t) 3,

y1(0) = 1 − ε, y2(0) = 0, y

1(0) = 0, y

2(0)=



1+ ε

1− ε , 0
t
20.

(4.5)

This problem can also be found in [16] or from the test set of problems in [22] The solution components are

y1(t)= cos
u(t)

− ε, y2(t)=(1 + ε)(1 − ε) sin
u(t)

, where u(t) is the solution of Keppler’s equation t = u(t) − ε sin(u(t)) and ε denotes the eccentricity of

the orbit In this example, we set ε = 0.9 The results reported in Table 2 show a similar efficiency of the

Table 1

Numerical results for problem (4.4)

EPTRKN8 method as for the Fehlberg problem when it is compared with ODEX2, DOPRIN and PIRKN codes

5 Concluding remarks

In this paper we have considered variable stepsize explicit pseudo two-step RKN methods requiring

only one effective sequential f -evaluation per step for any order of accuracy Implemented with a

variable stepsize strategy using embedding techniques, an explicit pseudo two-step RK methods derived from this class is shown to be superior to the currently most efficient sequential and parallel codes

as ODEX2, DOPRIN and PIRKN In a stringent accuracy range, these methods are expected to have

an efficiency equal if not superior to sequential codes even in a sequential implementation These conclusions encourage us to pursue the study of explicit pseudo two-step RKN methods In particular,

we will concentrate on the optimal choice of the method parameters, numerical experiments with higher-order explicit pseudo two-step RKN methods and also on an implementation of these methods on parallel computers

Table 2

Numerical results for problem (4.5)

Acknowledgements

A part of this work was done while I was a guest at the Institute of Numerical Mathematics, Halle University, Germany I would like to thank Prof Dr K Strehmel and Prof Dr R Weiner for their kind regards and interest in my research works I also would like to thank the referee for his/her useful comments which led to a further discussion concerning error estimates

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we will concentrate on the optimal choice of the method parameters, numerical experiments with higher-order explicit pseudo two-step. .. sequential f -evaluation per step for any order of accuracy Implemented with a

variable stepsize strategy using embedding techniques, an explicit pseudo two-step RK methods derived... compared with ODEX2, DOPRIN and PIRKN codes

5 Concluding remarks

In this paper we have considered variable stepsize explicit pseudo two-step RKN methods requiring

only