Some new implicit two-step multiderivativemethods for solving the special second-order

I.V.P. and the wave equation.

Higinio Ramos, M. F. Patricio

July 3, 2013

Abstract

The aim of this paper is to select from the huge family of implicittwo-step multiderivative methods those which have better character-istic concerning the error (order and error constant), stability andeasy of use (with most of the coefficients equal to zero). A few ofthese multiderivative methods which are A0-stable and A()-stablehave been obtained. This feature makes them useful for solving stiffproblems. The stability intervals or stability regions for the methodschosen are presented. Some numerical tests are included. The numer-ical test concerning the wave equation shows the good performanceof the methods of less order but A0-stable, compared with the poorperformance of the methods of higher orders.

1 Introduction

The numerical integration of the special second-order initial-value problemof the form

y(t) = f(t, y(t)) , y(t0) = y0 , y(t0) = y0 (1)

has received a lot of attention in the past decades as several physical problems(Schrodinger equations, orbital problems, electronics, molecular dynamics,semi-spatial discretization of wave equations, etc.) can be modelled by thisequation.

1

Two decades ago, several authors began to explore and develop multi-step Obrechkoff methods, originally introduced in [12] to integrate the prob-lem in (1) by using higher-order derivatives. There is some literature aboutObrechkoff methods for solving first-order initial-value problems ([8], [11],[1]), an also for second-order problems ([2], [13], [14]), although not veryextensive.

In [17] an analysis of P-stable symmetric two-step Obrechkoff methodsof orders 2m, m = 1, 2, . . . has been conducted, but for all the methodsconsidered there none of the coefficients is zero. In this paper we providesome methods with good stability characteristics, reasonable error and withas many coefficients as possible equal to zero. In the following sections wemade an analysis of the two-step multiderivative methods and present themethods with the best characteristics according to the stated criteria. Weapplied the obtained methods for solving a difficult problem obtained afterdiscretizing by the method of lines a wave partial differential equation. Thenumerical results show the good performance of the proposed methods.

2 Multiderivative multistep methods for the

special second-order IVP

Consider the general linear multiderivative multistep methods of the form

kj=0

j yn+j =

qi=1

h2ik

j=0

i j y(2i)n+j (2)

for solving the IVP

y = f(t, y) , y(t0) = y0 , y(t0) = y0

on a given interval [t0, tN ], where yn is an approximation to y(tn), tn =

t0 + nh, being h > 0 the constant stepsize, and y(2i)n+j an approximation for

the derivative y(2i)(tn+j) such that

y(2)(t) = f(t, y)

y(2i)(t) =d2y(2i2)(t)

dt2, i 2 .

2

Most of the definitions that follow may be found either in [4], [8] or [5].The order of the method is defined to be p if for an adequately smootharbitrary function z(t) it is

L[h, , ]z(t) = Cp+2 hp+2 z(p+2)(t) +O(hp+3) ,

with the error constant Cp+2 6= 0, where L[h, , ] is the linear operatordefined by

L[h, , ]z(t) =k

j=0

j z(t+ jh)qi=1

h2ik

j=0

i j z(2i)(t+ jh) .

Considering the usual Taylor series expansion it can be written as

L[h, , ]z(t) =p+1m=0

Cmhmz(m)(t) + Cp+2 h

p+2 z(p+2)(t) +O(hp+3) ,

where

C0 =k

j=0

j (3)

C1 =k

j=0

j j (4)

Cm =1

m!

kj=0

jm j ql=1

1

(m 2l)!k

j=0

l jjm2l ,m 2 (5)

and so, the conditions for the method to be of order p read

Cm = 0 , m = 0, 1, . . . , p+ 1 .

The coefficients {j}kj=0 and {i j} , i = 1, . . . , q ; j = 0, . . . , k can bedetermined imposing that L[h, , ]l(t) = 0 for a set of basis functions{l(t)}rl=0 , r k+(k+1)q. Traditionally, these basis functions are chosen tobe monomials, l(t) = t

l, but other choices are possible, obtaining in this case

3

the so-called adapted Obrechkoff methods (in [18], [16] exponentially-fittedsymmetric Obrechkoff methods are considered while in [20] a trigonometric-fitted method of twelfth order was presented).

Define (x) and i(x) to be the first and second characteristics polyno-mials given by

(x) =k

j=0

j xj , i(x) =

kj=0

i j xj , i = 1, . . . , q

which we assume that have no common factors. Using these polynomials themethod in (2) may be written as

(E) yn =

qi=1

h2ii(E) y(2i)n .

A method of the family in (2) is said to be zero-stable if no root of the firstcharacteristic polynomial (x) has modulus greater than one, and if everyroot of modulus one has multiplicity not greater than two.

The linear stability theory for these methods is related to the test equation

y(t) = y(t) .

The stability polynomial is defined by

(x, h) = (x)qi=1

hi i(x)

where h = h2 . A method of the family in (2) is said to be absolutely stablefor a given h if all the roots of (x, h) satisfy |ri| 1 , i = 1 . . . , k. The regionof the complex plane

D = {h C : the method is absolutely stable}is called stability domain or region of absolute stability of the method. Whenthe negative complex plane is included in D the method is said to be A-stable.

A method of the family in (2) is said to be A()-stable with 0 < pi/2,if the angular sector

S = {z C : | arg(z)| < , z 6= 0}

4

is contained in the stability domain D.An interval [a, b] of the real line is said to be an interval of absolute

stability if the method is absolutely stable for all h [a, b]. When theinterval of absolute stability consists in the negative real axis the method issaid to be A0-stable.

The method is convergent if and only if p 1 and the method is zero-stable (see Refs. [6, 7]).

The method is assumed to satisfy the following requirements:

k = 1 , |0|+qi=1

|i 0| 6= 0 , (normalization)

qi=0

i k 6= 0 (implicitness),

(x) and i(x) have no common factor (irreducibility), (1) = (1) = 0, [x(x)] (1) = 21(1) (consistency), the method is zero-stable (to assure convergence).

The method is called symmetric if

j = kj , i j = i kj , for i = 1, . . . , q ; j = 0, . . . , bk/2c.Particular cases of the methods in (2) are the Stormer-Cowell methods,

which are obtained for q = 1 and 0 = 1, 1 = 2, j = 0, j = 2, . . . , k 1.There are no methods of type (2) of one step, and thus two steps must beconsidered at least. Some examples of these methods are the method

yn 2yn+1 + yn+2 = h2y(2)nwhich is an explicit method with order p = 1, and the explicit method

yn 2yn+1 + yn+2 = h2y(2)n+1which has order p = 2 (this is an Stormer-Cowell method). The formula

yn 2yn+1 + yn+2 =qi=1

h2iy(2i)n+1

i (2i 1)!is a two step method of the form in (2) of order p = 2q with error constantC2q+2 = 1/((q + 1)(2q + 1)!) for solving (1) that may be considered as theequivalent to the Taylor method for first-order I.V.P.s. (the Taylor methodis a one-step explicit Obrechkoff method for first-order I.V.P.s [8]).

5

3 Analysis of two-step implicit Obrechkoff meth-

ods

In this section we are going to study the implicit Obrechkoff methods of theform in (2) for k = 2 and q = 1, 2, 3, and so the methods read

2j=0

j yn+j =

qi=1

h2i(i 0y

(2i)n + i 1y

(2i)n+1 + i 2y

(2i)n+2

). (6)

We will denote the methods byM(q, p) where q refers to the higher order 2q ofthe derivatives appearing in the method, and p indicates the algebraic orderof the method. We have only considered implicit methods due to their betterstability characteristics as we are interested in applying them for solving stiffproblems.

Our goal consists in looking for optimal methods of the form in (6)having the following properties:

1. high order with minimum absolute error constant

2. great stability regions or intervals of stability

3. reduced number of function evaluations, which means to have as muchas possible of the coefficients i j equal to zero

Taking 2 = 1 in order to normalize the parameters, and imposing C0 =C1 = 0 (necessary to guarantee the minimum order p = 1) results that thefirst characteristic polynomial is the same for all the methods, being

(x) = 1 2x+ x2 .

In what follows we will analyze different possibilities according to thevalue of q.

3.1 Methods with q=1:

Let us consider q = 1 in (6). The most known method of this type is theNumerov method [4] (the method named as M(1, 4) in Table 3.1), which hasthe highest order in this class.

6

There exist no values of the coefficients in (6) which cause the order tobe p = 3.

A one-parameter family of symmetric zero-stable methods of orderp = 2 whenever C4 6= 0 is given by

yn 2yn+1 + yn+2 = h2(1 0y

(2)n + (1 21 0)y(2)n+1 + 1 0y(2)n+2

)(7)

with error constant C4 =1121 0 (for 1 0 = 112 results the Numerov method).

For 1 0 14 the absolute stability interval is (, 0] (A0-stability), and thusour choice for this parameter will be on this range. For 1 0 =

14we obtain

the method

yn 2yn+1 + yn+2 = h2

4

(y(2)n + 2y

(2)n+1 + y

(2)n+2

)(8)

known as Dahlquist method (it is the method obtained using the (1, 1)-Padeapproximant in [22]), with error constant C4 =

16. Taking 1 0 =

12the

method in (7) reduces to

yn 2yn+1 + yn+2 = h2

2

(y(2)n + y

(2)n+2

), (9)

with error constant C4 =512.

A two-parameter family of zero-stable methods of order p = 1 when-ever C3 6= 0 is given by

yn 2yn+1 + yn+2 = h2((1 1 1 1 2)y(2)n + 1 1y(2)n+1 + 1 2y(2)n+2

)with error constant C3 = 1 1 1 21 2. For 1 1 = 1/2, 1 2 = 1/4 is C3 = 0and we obtain the method in (8) and for 1 1 = 0, 1 2 = 1/2 we obtain themethod in (9), which are second order methods. In fact, if C3 = 0,that is,1 1 = 1 21 2, we obtain the class of second-order methods in (7).

Taking 1 1 = 0, 1 2 = 1 arises the method

yn 2yn+1 + yn+2 = h2 y(2)n+2 , (10)

which has order p = 1. The absolute stability region for this method, thatincludes the negative real axis, is shown in Figure 1.

7

-2 0 2 4 6-4

-2

0

2

4

Figure 1: Absolute stability region for the method (10).

In Table 3.1 we include some of the characteristics of the optimal methodsin the family M(1, p).

Method p 1,0 1,1 1,2 Stability Cp+2

M(1, 4) 41

12

5

6

1

12[-6,0] 1

240

M(1, 2) 21

20

1

2A0 5

12M(1, 1) 1 0 0 1 A() -1

Table 1: Characteristics of the optimal methods in the family M(1, p).

3.2 Methods with q=2:

Now, let us consider q = 2 in (6). The method of higher order in this classhas appeared in [17] (the method named as M(2, 8) in Table 3.2) which has

8

p = 8, C10 =59

76204800, and stability interval [25.2, 0].

There are no values of the coefficients in (6) for which is obtained amethod of order p = 7.

A class of zero-stable symmetric methods of sixth order with a free pa-rameter a = 20 reads

yn 2yn+1 + yn+2 = h2((

1

30 12a

)(y(2)n + y

(2)n+2

)+

(24a+

14

15

)y(2)n+1

)

+h4(a(y(4)n + y

(4)n+2

)+

(10a+

1

20

)y(4)n+1

)with C8 = (13 + 15120a)/302400. We have found four methods of thisclass with non empty stability interval and with some of the coefficientsi j , i = 1, 2; j = 0, 1, 2 null:

1. for a = 7/180 the method has error constant C8 = 23/12096 andinterval of stability [6

47

(15 +

1165

), 0] ' [6.2721, 0]

yn 2yn+1 + yn+2 = h2

2

(y(2)n + y

(2)n+2

) h

4

180

(7y(4)n + 61y

(4)n+1 + 7y

(4)n+2

)2. for a = 1/200 the method has error constant C8 = 313/1512000 and

interval of stability [23

(1309 47) , 0] ' [7.2132, 0]

yn 2yn+1 + yn+2 = h2

75

(7y(2)n + 61y

(2)n+1 + 7y

(2)n+2

) h

4

200

(y(4)n + y

(4)n+2

)3. for a = 1/360 the method has error constant C8 = 11/60480 and

interval of stability [12, 0]

yn 2yn+1 + yn+2 = h2 y(2)n+1 +h4

360

(y(4)n + 28y

(4)n+1 + y

(4)n+2

)9

4. for a = 0 the method has error constant C8 =13

302400and interval of

stability [20, 0]

yn 2yn+1 + yn+2 = h2

30

(y(2)n + 28y

(2)n+1 + y

(2)n+2

)+h4

20y(4)n+1

There are no values of the coefficients in (6) for which the order is p = 5.A family of zero-stable symmetric fourth order methods with two free

parameters, a = 20 , b = 21, is given by

yn 2yn+1 + yn+2 = h2(1 24a b

12(y(2)n + y

(2)n+2) +

24a+ 12b+ 5

6y(2)n+1

)

+h4(ay(4)n + by

(4)n+1 + ay

(4)n+2

)with error constant C6 = (1 200a + 20b)/240. We have found threemethods of this class with non empty stability interval and with some of thecoefficients i j , i = 1, 2; j = 0, 1, 2 zero:

1. for a = 1/24 , b = 0 the method has error constant C6 =7180

and

interval of stability [2(321) , 0] ' [3.1651, 0]

yn 2yn+1 + yn+2 = h2 y(2)n+1 +h4

24

(y(4)n + y

(4)n+2

)2. for a = 0 , b = 5/12 the method has error constant C6 = 7

180and

interval of stability [25

(3 +

69), 0] ' [4.5226, 0]

yn 2yn+1 + yn+2 = h2

2

(y(2)n + y

(2)n+2

) 5h

4

12y(4)n+1

3. for a = 5/24 , b = 0 the method has error constant C6 = 61360

and

interval of stability (, 0]

yn 2yn+1 + yn+2 = h2

2

(y(2)n + y

(2)n+2

) h

4

24

(5y(4)n + 5y

(4)n+2

)10

There is a class of zero-stable symmetric methods of third order withthree free parameters, a = 20 , b = 21, c = 22 given by

yn 2yn+1 + yn+2 = h2

12

((1 d)

(y(2)n + y

(2)n+2

)+ 2(5 + d)y

(2)n+1

)+h4

(ay(4)n + by

(4)n+1 + cy

(4)n+2

)where d = 12(a+ b+ c), and with error constant C5 = a c. From this classwe only have considered the method for a = 0, b = 0, c = 5/12

yn 2yn+1 + yn+2 = h2

2

(y(2)n + y

(2)n+2

) h

4

125y

(4)n+2 (11)

which has the stability region shown in Figure 2.

-3 -2 -1 0 1 2 3-4

-2

0

2

4

Figure 2: Absolute stability region for the method (11).

In Table 3.2 the characteristics of the optimal methods of the familyM(2, p) have been included.

11

Method p 1,0 1,1 1,2 2,0 2,1 2,2 Stability Cp+2

M(2, 8) 811

252

230

252

11

252

1315120

626

15120

1315120

[25.2, 0] 5976204800

M(2, 6) 61

30

28

30

1

300

1

200 [20, 0] 13

302400

M(2, 4) 41

20

1

2

524

0524

A061

360

M(2, 3) 31

20

1

20 0

512

A()5

12

Table 2: Characteristics of the optimal methods in the family M(2, p).

3.3 Methods with q=3:

Finally, let us consider q = 3 in (6). The method of higher order in this class

has also appeared in [17] which has p = 12, C14 =45469

1697361329664000and

stability interval [9.7954, 0].Although the authors in [17] say that have no knowledge of any paper

in which this formula had been written down, this method had previouslyappeared in [23], [24], and at the same time in [15].

In [1, 17, 22] there is also a sixth order symmetric method of this class

with error constant C8 =1

50400and stability interval (, 0], but none of

the coefficients zero.There are no values of the coefficients in (6) for which the order is p = 11.We present now different classes of methods depending on some free pa-

rameters.There is a class of zero-stable symmetric methods of tenth order with a

free parameter a = 30 given by

12

yn 2yn+1 + yn+2 = h2(1 + 45360a

39

(y(2)n + y

(2)n+2

)+37 90720a

39y(2)n+1

)

+ h4(17 3326400a

65520

(y(4)n + y

(4)n+2

)+1907 34776000a

32760y(4)n+1

)

+ h6(a(y(6)n + y

(6)n+2

)+59 3155040a

65520y(6)n+1

)(12)

with C12 = (127 39251520a)/43589145600. We have considered differentvalues of the parameter a to get some of the coefficients zero. The results forthese values appear in Table 3, where we have included the error constantC12 and the stability intervals.

13

ja 0 1 2 C12 [, 0]

0 1 j139

3739

139

2.9135 109 [9.6275, 0]2 j

1765520

190732760

1765520

3 j 059

655200

593155040

1 j891878

850939

891878

1.3925 108 [28.6780, 0]2 j

19071577520

30257788760

19071577520

3 j59

31550400 59

3155040

173326400

1 j13660

317330

13660

7.5156 109 [9.4941, 0]2 j 0

7110

03 j

173326400

19071663200

173326400

190734776000

1 j6176900

28333450

6176900

4.6466 108 [26.7024, 0]2 j

72300

0 72300

3 j1907

347760003025717388000

190734776000

145360

1 j 0 1 0 2.2765 108 [9.2015, 0]2 j

1315120

6177560

1315120

3 j1

4536089

453601

45360

3790720

1 j12

0 12

3.6434 107 [25.4403, 0]2 j

31715120

28337560

31715120

3 j37

90720854536

3790720

Table 3: Some of the methods in (12) with order p = 10.

There are no values of the coefficients in (6) for which the order is p = 9.A class of zero-stable symmetric methods of eighth order depending on

two free parameters, a = 30, b = 31, is given in Table 4.

14

j 0 and 2 1

1 j50400a5040b+11

2525(10080a1008b23)

126

2 j282240a+10080b13

151201229760a+141120b+313

7560

3 j a b

Table 4: Coefficients for the class of eight-order methods with k = 2 , q = 3taking two free parameters.

We can choose different values for the parameters in order to have atleast two of the independent coefficients zero. There are 14 possibilities,from which some of them appear in Table 5. In particular, for a = b = 0 itresults the higher order method with q = 2 named as M(2, 8) in Table 3.2.

15

a jb 0 1 2 C10 [, 0]

120160

1 j 0 1 0 3.5824 106 [8.0792, 0]3

11202 j 0

112

03 j

120160

31120

120160

0 1 j 0 1 0 1.1022 106 [8.5893, 0]115040

2 j1

168023280

11680

3 j 0115040

01150400

1 j 0 1 0 9.8104 106 [11.3278, 0]0 2 j

92800

3234200

92800

3 j1150400

0 1150400

0 1 j12

0 12

2.0392 105 [20.4083, 0]231008

2 j9560

323840

9560

3 j 0231008

023

100801 j

12

0 12

9.3694 105 [7.3730, 0]0 2 j

731680

277840

731680

3 j23

100800 23

10080

0 1 j156

2728

156

3.3462 107 [9.1037, 0]13

100802 j 0

11168

03 j 0

1310080

013

2822401 j

27784

365392

27784

2.6812 106 [19.7195, 0]0 2 j 0

1152352

03 j

13282240

0 13282240

3131229760

1 j3233416

13851708

3233416

9.7634 106 [8.0133, 0]0 2 j

11520496

0 11520496

3 j313

12297600 313

1229760

Table 5: Some of the eight-order methods in Table (4) for different values ofthe parameters.

There are no values of the coefficients in (6) for which the order is p = 7.A class of zero-stable symmetric methods of sixth order with three free

parameters, a = 30, b = 31, c = 20, is given in Table 6.

16

j 0 and 2 1

1 j720a360b360c+1

302(360a+180b+180c+7)

15

2 j c480a+240b+200c+1

20

3 j a b

Table 6: Coefficients for the class of sixth-order methods with q = 3 takingthree free parameters.

From this family we have only considered the two methods that follow,with some of the coefficients zero and absolute stability interval (, 0]:

taking a = 1/720 , b = 0 , c = 0 we obtain the method

yn 2yn+1 + yn+2 = h2 y(2)n+1 +h4

12y(4)n+1 +

h6

720

(y(6)n + y

(6)n+2

)with C8 = 3/2240.

taking a = 61/720 , b = 0 , c = 5/24 we obtain the method

yn 2yn+1 + yn+2 = h2

2

(y(2)n + y

(2)n+2

) 5h

4

24

(y(4)n + y

(4)n+2

)+61h6

720

(y(6)n + y

(6)n+2

)with C8 = 277/4032.

Finally, a family of fifth order methods with four free parameters maybe considered. From this family we only take into account the zero-stablemethods with some of the coefficients zero and good regions of absolutestability (A()-stability). These methods are:

the method with error constant C7 = 1/360

yn 2yn+1 + yn+2 = h2 y(2)n+1 +h4

12y(4)n+1 +

h6

360y(6)n+2 (13)

17

the method with error constant C7 = 61/360

yn 2yn+1 + yn+2 = h2

2

(y(2)n + y

(2)n+2

) 5h

4

24

(y(4)n + y

(4)n+2

)+61h6

360y(6)n+2 (14)

The absolute stability regions of the methods (13) and (14) are plotted inFigure 3.

-20 -10 0 10 20 30 40 50-40

-20

0

20

40

-4 -2 0 2 4-4

-2

0

2

4

Figure 3: Absolute stability regions for the methods (13)[left] and (14)[right].

In Table 7 we have included the characteristics of the optimal methodsof the family M(3, p).

18

Method

p1,0

1,1

1,2

2,0

2,1

2,2

3,0

3,1

3,2

Stability

Cp+2

M(3,12)

12229

7788

7330

7788

229

7788

11

25960

1422

25960

11

25960

127

39251520

29230

39251520

127

39251520

[9.79,0]

45469

1697361329664000

M(3,10)

1089 1878

1700

1878

89 1878

1907

1577520

30257

1577520

1907

1577520

59

3155040

05

9

3155040

[28.67,0]

2923

209898501120

M(3,8)

811 252

230

252

11 252

13

15120

626

15120

13

15120

00

0[25.2,0]

59

76204800

M(3,6)

60

10

01 12

01 720

01 720

A0

5 12

M(3,5)

50

10

01 12

00

1 360

0A()

1 360

Table7:

Characteristicsoftheoptimalmethodsin

M(3,p).

4 Numerical time integration approach for

PDEs

For the integration of partial differential equations are used different tech-niques, among which are the finite element method, finite difference method,or the method of lines (MOL).

The latter procedure allows to obtain optimum results when the spatialdiscretization is fine enough and the temporal integrator has good propertiesof order, convergence and stability.

Let us consider the one-dimensional wave equation with initial and bound-ary conditions given by

2u(x, t)

t2= c22u(x, t) , x [0, L] , t [0, tf ]

u(x, t)

t(x, 0) = f1(x) , u(x, 0) = f2(x)

u(0, t) = g1(t) , u(L, t) = g2(t)

(15)

The MOL is a very powerful technique which has been widely used forthe solution of (15). The idea behind this approach consists in taking on thespace domain a discrete mesh

:= {xi [0, L] : 0 = x0 x1 . . . xN+1 = L} ,

not necessarily evenly spaced, in such a way that for every xi , the spatialderivatives appearing in (15) are approximated by means of finite differences,spectral methods or finite elements techniques.

Even though having a uniform step size between points makes it con-venient to write out the formulas, it is certainly not a requirement. On auniform step size mesh with step size4x = (xN+1x0)/(N+1) = L/(N+1),finite second order centered differences for the first and second derivatives ofa function (x) are commonly given by

(xi) =(xi+1) (xi1)

24x ,

(xi) =(xi+1) 2(xi) + (xi1)

(4x)2 . (16)

20

Thus, replacing the space derivative in (15) by the central difference approx-imation

2u(x, t)

x2' 1

(4x)2 [u(x+4x, t) 2u(x, t) + u(x4x, t)] ,

and setting ui(t) = u(xi, t) for i = 1, . . . , N , with values u0(t) = u(0, t) =g1(t) , uN+1(t) = u(L, t) = g2(t), from the problem in (15) we obtain theinitial-value problem of the form

d2 u

d t2= c2 (Au(t) +B(t)) ,

u(0) = (u1(0), . . . , uN(0))T ,

u(0) = (f1(x1), . . . , f1(xN))T

(17)

where u(t) = (u1(t), . . . , uN(t))T , the N N matrix A is

A =

2(4x)2

1(4x)2 0 . . . 0 0 0

1(4x)2

2(4x)2

1(4x)2 . . . 0 0 0

0 1(4x)2

2(4x)2 . . . 0 0 0

......

......

......

0 0 0 . . . 2(4x)2

1(4x)2 0

0 0 0 . . . 1(4x)2

2(4x)2

1(4x)2

0 0 0 . . . 0 1(4x)2

2(4x)2

,

and the N -vector B(t) is

B(t) =

(u0(t)

(4x)2 , 0, . . . , 0,uN+1(t)

(4x)2)T

=

(g1(t)

(4x)2 , 0, . . . , 0,g2(t)

(4x)2)T

.

Note that the eigenvalues of A are (see [21])

j =2

(4x)2(cos

(j pi

N + 1

) 1), j = 1, . . . , N ,

which belong to the range (4 (N +1)2, 0), and so, for large values of N thesystem becomes very stiff. We observe that for solving the problem (17) wemay use some appropriate method of the above section.

21

5 Numerical examples

We are going to apply the above methods for solving different problems inorder to see their performance. Firstly we include a scalar problem which willbe solved by two of the methods not A0-stable, and later a problem relatedwith the wave equation.

Note that the application of any of the methods in (6) for solving thesystem in (17) leads to a N N algebraic system, that must be solved. Forthat purpose any of the Newton-type methods may be used. In particular,for the problem (17) the resulting system is linear, which makes easier itsresolution.

5.1 Problem 1

Let us consider the initial-value problem in [15] given by

y(t) + 25y(t) = 8(cos(t) +

2

3cos(3t)

), y(0) = 1 , y(0) = 0 (18)

whose exact solution is

y(t) =1

3(cosx+ cos(3x) + cos(5x)) .

We have solved over five periods the above problem with the first of themethods in Table 3, which is a methodM(3, 10), and the sixth of the methodsin Table 5, which is a methodM(3, 8). These methods are given respectivelyby

yn 2yn+1 + yn+2 = h2

39

(y(2)n + 37y

(2)n+1 + y

(2)n+2

) h

4

65520

(17y(4)n 3814y(4)n+1 + 17y(4)n+2

)+

h6

65520

(59y

(6)n+1

)

22

M(3, 10) M(3, 8)h (1st in Table 3) (6th in Table 5)pi

108.22305 106 4.59128 104

pi

204.00879 109 1.35816 106

pi

403.71127 1012 2.76865 109

pi

801.75386 1015 5.22551 1012

Table 8: Maximum absolute errors for Problem 1.

and

yn 2yn+1 + yn+2 = h2

56

(y(2)n + 54y

(2)n+1 + y

(2)n+2

)+h4

168

(11y

(4)n+1

)+

h6

10080

(13y

(6)n+1

).

We observe that these methods have stability intervals very similar.In Table 8 appear the maximum absolute errors taking different values of

the stepsize h. The numerical solutions provided by the methods are quiteaccurate, even for relatively large values of h. In Figure 4 the plot of theexact solution and the discrete one obtained with the method M(3, 10) isshown (the plot with the method M(3, 8) is similar).

23

5 10 15 20 25 30

-1.0

-0.5

0.5

1.0

Figure 4: Exact solution (continuous line) and discrete solution in [0, 10pi]obtained with the method M(3, 10) taking h = pi/10 for the problem (18).

5.2 Problem 2

Let us consider the problem of the form in (15) given by

2u(x, t)

t2=2u(x, t)

x2, x [0, L] , t [0, tf ]

u(x, t)

t(x, 0) = sin(x) , u(x, 0) = 0

u(0, t) = 0 , u(L, t) = sin(L) sin(t)

(19)

whose exact solution is given by

u(x, t) = sin(x) sin(t) .

After discretizing according to the guidelines in the previous section we obtainthe differential system of the form in (17) with initial values

u1(0) = 0, . . . , uN(0) = 0; u1(0) = sin(x1), . . . , u

N(0) = sin(xN) .

24

M(1,2) M(2,4) M(3,6)tf L x = 10

1 x = 102 x = 101 x = 102 x = 101 x = 102

2 2 2.9 103 2.4 103 5.0 104 1.5 105 4.9 104 4.9 10610 4.3 103 3.6 103 7.3 104 2.1 105 7.2 104 7.2 10630 4.3 103 3.6 103 7.3 104 2.1 105 7.2 104 7.2 106

10 2 1.9 103 1.6 103 3.0 104 9.0 106 3.0 104 2.9 10610 2.3 102 1.9 102 4.0 103 1.2 104 3.9 103 3.9 10530 1.9 102 1.6 102 3.3 103 9.9 105 3.2 103 3.2 105

Table 9: Maximum absolute errors for the problem in (19) using differentoptimal methods.

We have solved this problem using some of the optimal two-step mul-tiderivative methods in Section 3 (see Tables 3.1, 3.2 and 7). As it wouldexpected, the methods that are not A0-stable are not appropriate for solvingthe problem, except if a very small stepsize 4t = h is taken. The methodsof higher order M(1, 4) (Numerov), M(2, 8) and M(3, 12) produce very bigerrors for the stepsizes considered. In Table 9 we have included the maximumabsolute errors with the optimal methods M(1, 2),M(2, 4) and M(3, 6) fordifferent values of L and tf taking x = 10

1, 102, 4t = 101.All the methods considered work well when there is a decrease in the

space stepsize 4x, and also when the time length increases, tf = 2, 10. As itwas expected, the best accuracy is obtained with the method M(3, 6), whichalso entails greater computational cost.

We have included in Fig. 5 the plots of the exact and discrete solutions[up], together with the two plots overlapped [down], showing the close agree-ment between them.

25

02

46

x

0

2

4

6

t-1

0

1

u

02

46

x

0

2

4

6

t-1

0

1

u

02

4

6

x

0

2

4

6

t

-1

0

1

u

Figure 5: Exact [up-left] and discrete [up-right] solutions of the problem (19)in [0, 2pi] [0, 2pi] using the method M(3, 6) taking 4x = pi/10,4t = 1/10.The two solutions overlapped [down].

6 Conclusions

We have made an analysis of the two-step multiderivative methods for solv-ing second order initial-value problems, looking for methods of the form in(6) with better characteristics concerning the error, the stability and lower

26

computational work. We have found that for q = 1, 2, 3, being 2q the highestorder of the derivative appearing in the method, there are methods A0-stablesof orders 2q, and methods A()-stables of orders 2q 1. These are the mostappropriate methods to be used in problems where stability is a requirement,such as stiff problems. On the contrary, the methods of high order, which arethe Numerov method namedM(1, 4) and the methodsM(2, 8) andM(3, 12),which had appeared in [18], in general are not suitable for solving such kindof problems.

The application of the methods M(1, 2),M(2, 4) and M(3, 6) presentedin the article to a highly oscillatory problem and to another concerning thewave equation shows the good performance of the proposed methods.

Acknowledgements

The first author is grateful for the hospitality during his stay at the Depart-ment of Mathematics, University of Coimbra, during which this work wascarried out.

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