A variable-stepsize variable-order multistep method for the integration of perturbed linear problems

Applied Numerical Mathematics 42 (2002) 285–295www.elsevier.com/locate/apnum

A variable-stepsize variable-order multistep methodfor the integration of perturbed linear problems✩

David J. Lópeza,∗, Pablo Martínb, Amelia Garcíab

a Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Paseo Alfonso XIII,30203 Cartagena, Spain

b Departamento de Matemática Aplicada a la Ingeniería, E.T.S. de Ingenieros Industriales, Universidad de Valladolid,Paseo del Cauce s/n, 47011 Valladolid, Spain

Abstract

In 1971 Scheifele wrote the solution of a perturbed oscillator as an expansion in terms of a new set of functions,which extends the monomials in the Taylor series of the solution. Recently, Martín and Ferrándiz constructed amultistep code based on the Scheifele technique, and it was generalized by López and Martín for perturbed linearproblems. However, the remarked codes are constant steplength methods, and efficient integrators must be able tochange the steplength. In this paper we extend the ideas of Krogh from Adams methods to the algorithm proposedby López and Martín, and we show the advantages of the new code in perturbed problems. 2001 IMACS.Published by Elsevier Science B.V. All rights reserved.

1. Introduction

We are interested in the numerical integration of problems of the kind

Lx = εg(x, x′, . . . , x(k−1), t

),

x(t0) = x(0)0 , x′(t0) = x

(1)0 , . . . , x(k−1)(t0) = x

(k−1)0 , (1)

where the derivatives are with respect tot , ε is a small parameter andL is the linear operator

Lx = x(k) + λ1x(k−1) + · · · + λkx. (2)

✩ This work has been partially supported byJunta de Castilla y León under project VA11/99, by the SpanishDGES underproject PB95-0696, by the SpanishMinisterio de Ciencia y Tecnología under project AYA2000-1787 and the Comunidad deMurcia under project PL-53/00809 IFS/01.

* Corresponding author.E-mail addresses: [email protected] (D.J. López), [email protected] (P. Martín), [email protected] (A. García).

0168-9274/01/$22.00 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.PII: S0168-9274(01)00156-8

286 D.J. López et al. / Applied Numerical Mathematics 42 (2002) 285–295

Classical codes, such as Runge–Kutta or Adams methods, require a formulation of the problem as afirst order ODE and start from the Taylor expansion of the solutionx and itsk − 1 first derivatives, sothey do not take advantage of the magnitude ofε. In opposition to these codes, the algorithm proposedby López and Martín [12] starts from the Taylor series of the perturbation termεg, which makes thesmall parameterε appear as a common factor of the error. This fact does not hold when problem (1) isintegrated with a classical code.

The aim of this work is the extension of the aforementioned algorithm of López and Martín to avariable stepsize code. In the second section we will give a brief exposition of the algorithm, withspecial attention to the implementation mode of the method. In the third one we will extend the Kroghcoefficients for the Adams method to this code, and we will develop a variable-stepsize code adequate tothe integration of (1), obtaining the Adams, Falkner [3] and Martín and Ferrándiz [13] variable steplengthalgorithms as particular cases. Finally, we will present some numerical examples that will show theadvantages of the new code when integrating perturbed linear problems.

2. The algorithm of López and Martín

We defineHn, 0� n < k, as the solutions of the homogeneous problems

LHn = 0, H (i)n (0) = 0, 0 � i < k, i �= n, H (n)

n (0) = 1. (3)

Then, assuming regularity tog, the solutionx and itsk − 1 derivatives are given by the equality

x(i)(t0 + h) =k−1∑n=0

x(n)(t0)H(i)n (h)+ ε

t0+h∫t0

G(i)(t0 + h− s)g(s)ds, (4)

whereG = Hk−1 is the Green function associated toL. The method of López and Martín (LM method)[12] is obtained with the substitution ofg with an interpolating polynomial in (4). The scheme is

x(i)

m+1 =k−1∑n=0

x(n)m H (i)n (h)+ εhk−i

r−1∑n=0

β(i)n,σ (h)∇ngm+σ , 0 � i < k, (5)

wheregm+σ = g(x(0)m+σ , . . . , x

(k−1)m+σ , tm+σ ) with σ = 0 for the explicit method andσ = 1 for the implicit

one. The coefficients can be obtained from

β(i)n,σ (h) = (−1)n

hk−i

h∫0

G(i)(h− s)

(−s/h+ σ

n

)ds. (6)

2.1. Properties of the LM algorithm

2.1.1. ConvergenceIt follows from the interpolating construction of the algorithm that formula (5) isr th order consistent,

and it is exact wheng is a polynomial of degree less thanr .From Grigorieff [7], to study the stability we must rewrite (5) as an one-step algorithm and prove

that the set made by the powers of an exponential matrix (associated to the coefficients of the linear

D.J. López et al. / Applied Numerical Mathematics 42 (2002) 285–295 287

operatorL) is bounded. The proof is similar to Martín and Ferrándiz [13], and still holds for variablestepsizes.

Then the algorithm (5) isr th order convergent and, as the unperturbed problem is the case wheng isthe null polynomial, it is exact whenε = 0.

2.1.2. Particular casesAs the LM formulae are exact wheng is a polynomial, it is easy to see that they generalize to the

Adams methods for the first order operatorLx = x′, to the Falkner ones forLx = x′′ and to the Martínand Ferrándiz codes forLx = x′′ + λx. It is reasonable to investigate the variable stepsize extension ofthese methods starting from the classical Krogh [9] coefficients. We must go ahead from the examinationof the predictor–corrector pair.

2.1.3. The predictor–corrector pairWe will show in this subsection how Milne’s device to estimate the local error can be applied even in

the case of the LM algorithm. Now we study the local extrapolation in a pair based onr th order explicitand implicit methods. It follows from (6) that the property

β(i)

n+1,1(h)= β(i)

n+1,0(h)− β(i)

n,0(h), 0� i < k, n� 0, (7)

holds, which implies that

ε

r−1∑n=0

(β(i)n,1(h)∇ngm+1 − β

(i)n,0(h)∇ngm

)= εβ(i)r−1,0(h)∇rgm+1, 0� i < k. (8)

Eq. (8) is essential for the behaviour of the local extrapolation (see Lambert [10, pp. 110–115]). An easycomputation shows that the Milne’s estimate for the principal part of the local truncation error (PLTE)can be extended to

PLTE(i)

m+1 εβ(i)r,1(h)

β(i)r,0(h)− β

(i)r,1(h)

(x(i)[µ]m+1 − x

(i)[0]m+1

), (9)

so ther th order pair with local extrapolation is equivalent to ther-steps pair (r th order explicit predictor,(r + 1)th order implicit corrector). The algorithm is

(1) Predictor:x(i)[0]m+1 =∑k−1

n=0 x(n)m H (i)

n (h)+ εhk−i∑r−1

n=0β(i)

n,0(h)∇ngm.

(2) First corrector:x(i)[1]m+1 = x

(i)[0]m+1 + εhk−iβ

(i)r,0(h)∇rg

[0]m+1.

(3) Other correctors (if needed):x(i)[ν+1]m+1 = x

(i)[ν]m+1 + εhk−iβ

(i)r,0(h)(g

[ν]m+1 − g

[ν−1]m+1 ).

2.2. Implementation mode

Let us consider the test problemx′′ + 100x = −x/4t2 in [t0, tf ] = [1,10]. The initial values arechosen from the exact solutionx(t) = √

tJ0(10t), whereJ0 is the null-order first-kind Bessel function.We present the results of the implicit fourth order LM method with the linear operatorLx = x′′ + 100x.The error has been taken as the maximum error of solution and derivative att = 10 for several predictor–corrector implementations, withh = 1/100.


Table 1Implicit LM method. Solution, derivative and error int = 10

LM x x′ Error

P(EC) 0.0632008138924442 2.4427102352387662 3.7759× 10−8

P(EC)E 0.0632008138912906 2.4427102352247765 3.7773× 10−8

P(EC)2 0.0632008138905629 2.4427102352166949 3.7781× 10−8

P(EC)2E 0.0632008138905629 2.4427102352166949 3.7781× 10−8

The table shows that, for the type of problems under consideration,P(EC) seems to be the mosteffective way to implement the LM method. This is due to the small quantity that we are adding in thecorrector stage (notice the small parameterε). We recommend to the reader the numerical experimenty′ = εy3, y(0) = 1, in order to test the ABM pair whenε is small: all the implementations produce veryclose results and the most efficient one is theP(EC) mode. This fact is not true for general IVPs.

3. Variable coefficient techniques

The best general reference here is Hairer et al. [8, pp. 397–400]. Our variable stepsize algorithm canalso be defined by substitutingg with an unevenly spaced interpolating polynomial in (4),

x(i)m+1 =

k−1∑n=0

x(n)m H (i)n (hm)+ εhk−i

m

r−1∑n=0

βn(m)γ (i)n (m)φn(m), (10)

and it can be expressed in terms of the three sets of coefficients

βn(m) =n−1∏j=0

tm+1 − tm−j

tm − tm−j−1,

γ (i)n (m) = 1

hk−im

tm+1∫tm

G(i)(tm+1 − s)

n−1∏j=0

s − tm−j

tm+1 − tm−j

ds,

φn(m)= g[tm, . . . , tm−n]n−1∏j=0

(tm − tm−j−1). (11)

The valuesβn(m) andφn(m) are the known Krogh’s coefficients, and can be calculated by the classicalalgorithms:

β0(m)= 1, βn(m) = βn−1(m)tm+1 − tm−n+1

tm − tm−n

,

φ0(m) = gm, φn(m)= φn−1(m)− βn−1(m− 1)φn−1(m− 1). (12)

Theγ (i)n (m) coefficients are related togj in [8], but here they also depend on the Green function. Let the

auxiliary variablesc(i)n,q be given by


c(i)n,q = (q − 1)!hqm

tm+1∫tm

ξq−1∫tm

· · ·ξ1∫

tm

G(i)(tm+1 − ξ0)

hk−i−1m

n−1∏j=0

ξ0 − tm−j

tm+1 − tm−j

dξ0 · · ·dξq−1. (13)

Then,γ (i)n (m) = c

(i)n,1, and

c(i)n,q = c(i)n−1,q − hm

tm+1 − tm−n+1c(i)n−1,q+1. (14)

The proof of this item can be easily adapted from [8, Lemma 5.1]. A difference with the Adamscoefficientsgj in Hairer et al. [8] arises in the computation of the starting valuesc

(0)0,q . In the following

subsections we explain the way to calculate these coefficients.

3.1. Computing the starting values c(i)0,q

3.1.1. Taylor seriesAs a previous step to calculate the powers expansion ofc

(i)0,q we need to check the following lemma:

Lemma 1.Let

ωn,q = (q − 1)!hqm

tm+1∫tm

ξq−1∫tm

· · ·ξ1∫

tm

(tm+1 − ξ0)n

hnmdξ0 · · ·dξq−1.

Then we have ωn,q = 1q+n

.

Proof. Starting from the relation

(tm+1 − ξ0)n =

n∑j=0

(n

j

)hjm(−1)n−j (ξ0 − tm)

n−j , (15)

it follows that

ωn,q = (q − 1)!hq+nm

n∑j=0

(−1)n−j

(n

j

)hjmh

n−j+qm

(n− j)!(n− j + q)!

= 1

q + n

n∑j=0

(−1)n−j

(n+q

j

)(n+q−1

n

) = 1

q + n

n∑j=0

(−1)n−j

(n+q−1

j

)+ (n+q−1j−1

)(n+q−1

n

)= 1

q + n

(n+q−1

n

)+ (−1)n(n+q−1

−1

)(n+q−1

n

) = 1

q + n. ✷ (16)

The coefficientsc(i)0,q can now be written in terms of the power expansion of the Green functionG(t) = Hk−1(t). This function is O(tk−1), ∀t ∈ R it coincides with its power series centered int = 0and it is not complicated to develop a recurrence relation to writeG(n+k−1)(0) in terms ofλ1, . . . , λk, thecoefficients ofL (López [11]). Applying the Lemma 1 to (13) withn = 0 we obtain the Taylor series ofthe starting coefficients


c(i)0,q = (q − 1)!

hq+k−i−1m

∞∑n=k−i−1

tm+1∫tm

ξq−1∫tm

· · ·ξ1∫

tm

G(n+i)(0)(tm+1 − ξ0)

n

n!

= 1

hk−i−1m

∞∑n=k−i−1

G(n+i)(0)ωn,q

n! hnm

=∞∑n=0

G(n+k−1)(0)

(q + n+ k − i − 1) · (n+ k − i − 1)!hnm. (17)

3.1.2. Simplifying the starting values: relation in terms of q + j

The computation of the series (17) is a delicate point, and it will seriously increase the computationalcost of the method. We look for a relation ofc

(i)0,q in terms ofc(i)0,q+j .

Theorem 2.The starting values c(i)0,q satisfy the recursive relation

c(i)0,q =

k∑j=1

(−1)j−1λjhjm

(q − 1)!(q + j − 1)!c

(i)0,q+j +

k−1∑j=0

(−1)k−j−1hi−jm

(q − 1)!(q + k − j − 1)!H

(i)j (hm).

Proof. We give a brief idea of the demonstration. We need to define

G(−n)(t) =t∫

0

G−(n−1)(s)ds (18)

starting from G(0) = G = Hk−1. These functions satisfyG(−n) = Gn+k−1, where Gn+k−1 are thegeneralization of the second orderG-functions (Scheifele [14], Stiefel and Scheifele [15, pp. 141–150])made by López and Martín in [12]. The functionG(tm+1 − ξ0) = G(hm + (tm − ξ0)) in (13) is a solution(as function of the stepsizehm) of the homogeneous problemLx = 0. Then we obtain

G(i)(tm+1 − ξ0) =k−1∑j=0

G(j)(tm − ξ0)H(i)j (hm). (19)

The use of (19) in (13) whenn = 0 and aq-times successive integration lead us to the relation

c(i)

0,q = (−1)q(q − 1)!hq+k−i−1m

k−1∑j=0

G(j−q)(−hm)H(i)j (hm). (20)

As G(−n), n > 0, is the solution of the problem

Lx = tn−1

(n− 1)! , x(0) = · · · = x(k−1)(0) = 0, (21)

(see [12, Eq. (5)]) it is easy to check that

G(j−q)(t) = −k∑

l=1

λlG(j−q−l)(t)+ tq+k−j−1

(q + k − j − 1)! . (22)


Replacing (22) in (20) we obtain

c(i)

0,q = (−1)q(q − 1)!hq+k−i−1m

k−1∑j=0

(−

k∑l=1

λlG(j−q−l)(−hm)

)H

(i)j (hm)

+k−1∑j=0

(−1)k−j−1hi−jm

(q − 1)!(q + k − j − 1)!H

(i)j (hm), (23)

and the theorem is proved by changing the sum order and applying (20) in the last equality.✷3.1.3. Simplifying the starting values: relation in terms of i + 1Theorem 3.The starting values satisfy the relation

c(i)

0,q = 1

q

(G(i)(hm)

hk−i−1m

− c(i+1)0,q+1

).

Proof. The result immediately follows from

ξ1∫tm

G(i)(tm+1 − ξ0)dξ0 = −G(i−1)(tm+1 − ξ1)+G(i−1)(hm). ✷ (24)

3.1.4. Computation of c(i)0,q

The computation will follow the algorithm:

(1) Calculatec(k−1)0,q , r + 1� q � r + k, truncating its power series (17).

(2) Use the recursive relation in Theorem 2 withi = k − 1 to findc(k−1)0,q , r � q � 1.

(3) Computec(i)0,q , k − 1> i � 0, from Theorem 3.

We have not a criterion to decide how many terms must been taken in (1) to approximate the seriesfor every operatorL and every computer precision. We must remark that (2) increases the order ofaccuracy toc(k−1)

0,q , 1� q � r . Now we present the preceding algorithm for the most important particularcase of our problems, the perturbed oscillator. LetLx = x′′ + ω2x, thenH0(t) = cos(ωt) = G′(t),H1(t) = sin(ωt)/ω =G(t). First of all, we truncate the series

c(1)0,r+1 =

∞∑n=0

(−1)n(ωhm)

2n

(2n+ r + 1)(2n)! ,

c(1)0,r+2 =

∞∑n=0

(−1)n(ωhm)

2n

(2n+ r + 2)(2n)! (25)

(the degree of the polynomial approximation depends onω, the machine precision and the maximumallowed stepsize) to calculate this couple of values from the Horner algorithm. The rest of the coefficients(r � q � 1) for the derivative of the solution are taken from


c(1)0,q = − (ωhm)

2

(q + 1)qc(1)0,q+2 + 1

q

(cos(ωhm)+ ωhsin(ωhm)

q + 1

). (26)

For i = 0 we apply the algorithm

c(0)0,q = 1

q

(sin(ωhm)

ωhm− c

(1)0,q+1

), (27)

to find the starting values with 1� q � r + 1 associated to the solution.

4. A numerical example

In this section we integrate the orbit of an artificial satellite in focal variables [5,6]. The first threeoscillators in [13, Eq. (37)] are trivial to solve when we chose an equatorial satellite. The fourth one is

u′′ = −u+ 1+ 12J2u2

H(1+ e),

u(π) = 1− e

H(1+ e), u′(π)= 0, (28)

whereJ2 = 5× 10−4 is the perturbation term due to the Earth oblateness,e ∈ [0,1) is the eccentricity ofthe orbit andH > 1 is the height in the perigee (measured in Earth-radii). The artificial satellite equationsare an important test for multistep methods, since a more realistic model includes several thousands ofperturbation terms, so the reduction of the number of the function evaluations is crucial. Notice that theform of the problem (28) does not conform to the type we are considering. Anyway, it is easy to see thatLM methods can be applied to problems of the formLx = P(t)+ εg, whereP is a polynomial of degreeless than the order of the method, obtaining the same properties as when applied to problem (1). TheLM algorithm can be successfully applied to other kind of problems, like the example in Table 1. In thisexample there is not a perturbation factorε, but limt→∞ |g| = 0.

We have integrated in double precision a medium eccentric(e = 0.5) low Earth(H = 1.05) satellitewith three particular cases of the variable stepsize LM algorithm: ABM and Falkner inP(EC)E mode,and the code of Martín and Ferrándiz (SMF), that is, LM with the operatorLx = x′′ +x, inP(EC) mode.We compare the multistep methods with the popular explicit eight order Runge–Kutta–Nystrom methodDOPRIN86 [1,2]. The theoretical solution has been approximated following the techniques described inFarto et al. [4].

All the algorithms have been programmed by the authors, with the same peculiarities: initial stepsizeh0 = TOL, maximum allowed stepsizehmax = π and security conditionhm+1 � 5hm. In the multistepmethods we have used, for no particular reason, ar-order pair with local extrapolation (r-steps formula),so the corrector termεhk−i

m γ (i)r φr(m+ 1) is added to the predictor (10). Then the error estimation for the

ith derivative of the formula is

LE(i)r = εhk−i

m φr(m+ 1)(γ (i)r (m)− γ

(i)r−1(m)

). (29)

We have chosen the infinite norm to define

ESTr = max{∥∥LE(i)

r

∥∥∞, 0 � i < k}. (30)


If EST r > TOL we reduce the stepsize tohm = 0.7hm(TOL/EST r)1/(r+1), and we do not let the steplength

increase in the next accepted step. On the other hand, if the step is successful, we define the estimationsof the(r − 1)-steps and(r + 1)-steps formulae replacingr with r − 1 andr + 1 in (30). The new order isthe one with the smallest error estimation, and it can vary between 2 and 13(1 � r � 12). For all methodsthe new stepsizehm+1 uses a security factor 0.85 instead of the factor 0.7 for the rejections.

Figs. 1 and 2 show the maximum error between the solution and the derivative after 1000 revolutions.Fig. 1 shows that the SMF code, the multistep method with the most reasonable selection of the linearoperatorL, is the best of the algorithms when we compare the error and the function evaluations. In thatfigure we can see that all the multistep codes perform better than DOPRIN86, as a result of their lownumber of evaluations per step and their high orders: the slope of the regression lines are−12.7546 forthe SMF,−12.5468 for the Falkner and−12.9546 for the Adams, and only−7.56706 for the eighthorder DOPRIN86.

In Fig. 2 we have studied the CPU time (measured in uclocks, about 840 nanoseconds), and we observethat the difference between the Runge–Kutta–Nystrom and the multistep methods is immense in ourproblem, whereas our variable stepsize SMF method is slightly better than the other linear codes.

To estimate the results in a more realistic satellite we must increase the perturbation number, butin double precision only the first terms produces any significant results, because any value smallerthan 10−15 does not alter the numerical solutions. We have solved this difficulty by integrating 10revolutions of the problem withJ2, but now forcing to the codes to evaluate 1000 times the function

Fig. 1. 1000 revolutions. PerturbationJ2. Error vs evaluations.

Fig. 2. 1000 revolutions. PerturbationJ2. Error vs CPU time.


Fig. 3. 10 revolutions. PerturbationJ2 1000 times. Error vs CPU time.

(1 + 12J2u2)/(H(1+ e)) each time they need to calculate it. It is expected that the results in a satellite

with 1000 of different perturbation terms will be close to Fig. 3 (the results comparing the number ofevaluations vs CPU time are identical to this figure). We can see that the best method is the LM codewith the natural choice ofL, and the results are similar to Fig. 1 as we presuppose. Now the importance ofthe selection of the linear operator is clear when we integrate with a variable stepsize method a perturbedproblem with a function hard to evaluate.

Acknowledgements

The authors thank the referees, whose comments contributed to the improvement of the paper.

References

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A variable-stepsize variable-order multistep method for the integration of perturbed linear problems