A class of a stable order four and six linear multistep methods for stiff initial value problems

Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013

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A Class of A-Stable Order Four and Six Linear Multistep
Methods for Stiff Initial Value Problems
Kumleng G.M1, Longwap S1, Adee S. O2
1. Department of Mathematics University of Jos, P.M.B. 2084, Plateau State, Nigeria
2. Department of Mathematics, Fed. University of Tech. Yola, Nigeria
* E-mail of the corresponding author: kumleng_g@yahoo.com
Abstract
A new three and five step block linear methods based on the Adams family for the direct solution of stiff initial
value problems (IVPs) are proposed. The main methods together with the additional methods which constitute
the block methods are derived via interpolation and collocation procedures. These methods are of uniform order
four and six for the three and five step methods respectively. The stability analysis of the two methods indicates
that the methods are A–stable, consistent and zero stable. Numerical results obtained using the proposed new
block methods show that they are attractive for the solutions of stiff problems and compete favorably with the
well-known Matlab stiff ODE solver ODE23S.
Keywords: Linear multistep methods, initial value problems, interpolation and collocation.
1. Introduction
In this paper, we shall be concerned with the approximate numerical integration of the stiff initial value problem

y ' = f ( x, y ), y ( x0 ) = y0 , x ∈ [ a, b], y ∈ R

(1)
Numerical analysts have focused most of their works on the development of more efficient and accurate methods
for the solution of stiff problems, and as such many methods have been proposed. Notable among these methods
is the Backward Differentiation formulae (BDF), because of its wide region of absolute stability. Several
researchers such as Kim (2010), Chollom et. al. (2012), Kumleng et.al (2011, 2012) have proposed methods for
the numerical solution of (1) which were shown to be very efficient.
In this paper, we describe the construction of a new three and five step methods based on the Adams
methods similar to the generalized Adams methods of Brugnano and Trigiante (1998). These methods are
constructed using the interpolation concept where the continuous interpolants provide the block methods through
evaluation at some grid points. This approach produce two self- starting new A – stable block methods which
provide the solutions of stiff initial value problems on non-overlapping intervals.
2. The New Method
In this section, the new A – stable block methods shall be constructed based on the continuous finite difference
approximation approach using the interpolation and collocation criteria described by Lie and Norsett (1981)
called multistep collocation (MC) and block multistep methods by Onumanyi et al. (1994).
We define based on the interpolation and collocation methods the continuous form of the k- step new
method as
m

yn + v − α v −1 ( x) yn+ v −1 = h∑ β j ( x) f n+ j
j =0

where

α v−1 and β j ( x)

(2)

are the continuous coefficients of the method, m is the number of distinct collocation

points, h is the step size and

v=

k −1
2

, k = 3,5, 7...

From Onumanyi et. al (1994), we obtain our matrices D and C = D

1 xn+ v −1

1
0
D = 0
1

M
0
1


2
xn+ v −1 L

2 x0
2 x1

L
L

2 xm −1

L
L

m
xn + v −1 
m 
mx0 −1 
mx1m −1 

M 
m −1
mxm −1 


−1

as

(3)

and
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Vol.3, No.11, 2013

 α v −1,1
α
v −1,2
C=
 M

α v −1,m +1

hβ 0,1
hβ 0,2
hβ 0,m +1

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hβ m−1,1 
hβ m −1,2 


M

L hβ m−1,m+1 
L
L

(4)

respectively.
2.1 Derivation of the Three Step New Block Method
In this case, k = 3, v=1, t = 1 and m = 4 and its continuous form expressed in the form of (2) is;

y ( x ) = α1 ( x ) yn + h ( β 0 ( x) f n + β1 ( x ) f n +1 + β 2 ( x) f n + 2 + β3 ( x ) f n + 3 )

(5)

The matrix D, in (3) becomes

 1 xn

0 1
D = 0 1

0 1
0 1


2
xn

3
xn

2 xn
2 xn+1

2
3 xn
2
3 xn +1

2 xn + 2
2 xn +3

2
3 xn + 2
2
3 xn +3

4
xn 
3 
4 xn 
3
4 xn +1 

3
4 xn + 2 
3 
4 xn +3 

(6)

Using the Maple software, the inverse of the matrix in (6) is obtained and this yields the elements of the matrix C.
The elements of the matrix C substituted into (5) yields the continuous formulation of the method as:

(
+ (−

)

λ
y( xn + λ ) = yn + −λ − 11λh + 3λh2 − 24 h3 f n +
12
2

3λ
4h

2

3

4

)

+ 2λ2 − 8λh3 f n+ 2 +
3h
3

Evaluating (7) at the following points λ = h,
constitute the new three step block method.

4

(

λ

2

6h

(

3λ 2
2h

)

− 5λ2 + 8λh3 f n +1
6h
3

4

)

λ
− 6λh2 + 24 h3 f n +3
3

4

λ = 2h, λ = 3h yields the following discrete methods which

( 9 f n + 19 f n +1 − 5 f n + 2 + f n +3 )
yn + 2 − yn = h ( f n + 4 f n +1 + f n + 2 )
3
h
y n + 3 − y n = 8 ( 3 f n + 9 f n +1 + 9 f n + 2 + 3 f n + 3 )
yn +1 − yn =

(7)

h
24

(8)

The main new discrete scheme for the three step method is

yn +1 − yn =

h
24

( 9 f n + 19 f n +1 − 5 f n +2 + f n +3 )

(9)
This new method is consistent since its order is 4, it is also zero-stable, above all, it is A – stable as can be seen
in figure 1. The new three step discrete methods that constitute the block method (8) have the following orders
and error constants as can be seen in Table 1.
2.2 Derivation of the Five –Step New Block Method
In this case, k = 5,
t = 1 and m = 6 and its continuous form expressed in the form of (2) is;

y ( x ) = α1 ( x ) yn +1 + h ( β 0 ( x ) f n + β1 ( x ) f n +1 + β 2 ( x ) f n + 2 + β 3 ( x ) f n + 3 + β 4 ( x ) f n + 4 + β 5 ( x ) f n + 5 )

Similarly, we generate the continuous formulation of the new five step method as;

2

(10)

Vol.3, No.11, 2013

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(
+ (− h + −
+
−
+( h− +
−
+
+(− h + −
+
−
+( h − +
−
+
+(− h + −
+
−

)

95
λ
λ
y( xn + λ ) = yn+1 + − 288 h + λ − 137λh + 5λ2 − 17λ3 + 40h4 − 720h5 f n
120
8h
96 h
2

5λ 2
2h

1427
1440

5λ 2
2h

133
240

173
1440

77 λ 3
36 h2

5λ
8h

3
160

71λ 4
96 h3

13λ 5
60 h4

13λ 3
6 h2

49λ 4
48 h3

5 h4

3

λ2

5λ 3
36 h2

10 h

41λ
96 h3

4

7λ4
90 h3

5

6

6

6

6

11λ
120 h4

λ
− 144 h5

λ5

λ
+ 720h5

5

60 h4

)
)f
)f
)f

λ
+ 144h5 f n+2
λ
+ 720h5

λ5

)

λ
+ 144h5 fn +1

7 λ5
60 h4

59λ 4
48h3

61λ
72 h2

2

4

107 λ 3
36 h2

5λ 2
3h

241
720

3

6

6

n +3

n +4
n +5

(11)

Evaluating (11) at the following points λ = 0, λ = 2 h, λ = 3h, λ = 4 h, λ = 5h yields the following
discrete methods which constitute the new five step block method.
h
yn + 1 - yn = 1440 ( 475 fn + 1427 fn + 1 - 798 fn + 2 + 482 fn + 3 -173fn + 4 + 27 fn + 5 )
h
yn + 2 - yn+1 = 1440 ( -27 fn + 637 fn + 1 +1022 fn + 2 -258 fn + 3 +77 fn + 4 -11 fn + 5 )
h
yn + 3 - yn+1 = 90 ( - fn + 34 fn + 1 +114 fn + 2 +34 fn + 3 - fn + 4 )
h
yn + 4 - yn+1 = 160 ( -3fn + 69 fn + 1 +174 fn + 2 +174 fn + 3 +69 fn + 4 -3 fn + 5 )
h
yn + 5 - yn+1 = 45 (14 fn + 1 +64 fn + 2 +24 fn + 3 +64 fn + 4 +14 fn + 5 )

(12)

The main new discrete scheme for the five step method is
h
yn + 2 - yn +1 = 1440 ( −27 f n + 637 f n + 1 + 1022 f n + 2 − 258 f n + 3 + 77 f n + 4 − 11 f n + 5 )

(13)
This new method is consistent since its order is 6, its also zero-stable, above all, it is A – stable as can be seen in
figure 1.
The new five step discrete methods (12) have the following orders and error constants as shown in Table 1.
3. Analysis of the New Methods
In this section, we consider the analysis of the newly constructed methods. Their convergence is determined and
their regions of absolute stability are plotted.
3.1 Convergence
The convergence of the new block methods is determined using the approach by Fatunla (1991) and Chollom et.
al (2007) for linear multistep methods, where the block methods are represented in a single block, r point
multistep method of the form
k

k

i =1

i =0

A(0) ym +1 = ∑ A(i ) ym+1 + h∑ B (i ) f m −1
(14)

where h is a fixed mesh size within a block, A , B , i = 0,1, 2,...k are
i

ym , ym −1 and ym +1

i

r × r identity matrix while

are vectors of numerical estimates.

Definition 1: A block method is zero stable provided the roots

polynomial ρ ( R ) specified as



k

Rij = 1, 2,...k

of the first characteristic



ρ ( R) = det ∑ A(i ) R k −i  = 0
 i =0

satisfies

Rj ≤ 1



(15)

, the multiplicity must not exceed two. Fatunla (1994)
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Vol.3, No.11, 2013

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The block method (8) expressed in the form of (14) gives

 19
 1 0 0   yn +1   0 0 1   yn − 2 
24
 4
 




 0 1 0   yn + 2  =  0 0 1   yn −1  + h  3
 0 0 1   y   0 0 1  y 
 9

  n +3  
 n 
 8

−5
24

  f n +1   0 0
 

0   fn+2  +  0 0
3 
 0 0
8   f n +3 


1
24

1
3
9
8

9
24
1
3
3
8

  f n−2  


  f n −1  


  fn 


(16)

where
−5
9
1
 19 24 24 
 0 0 24 
1 0 0
 0 0 1
24








A(0) =  0 1 0  , A(1) =  0 0 1 , B (0) =  4 1 0  , B (1) =  0 0 1 
3
3
3
0 0 1
 0 0 1
3 
9 9 3





8 8 8
0 0 8 
(0)
(1)
Substituting A and A into (15) gives the characteristic polynomial of the block method (8) as
ρ (λ ) = det(λ A(0) − A(1) )

  1 0 0   0 0 1 
 
 

= det λ  0 1 0  −  0 0 1 
 

  0 0 1   0 0 1 
 

 λ 0 −1 
= det  0 λ −1 


 0 0 λ − 1


= λ 2 (λ − 1) = 0
λ = 1, λ2 = λ3 = 0 . The block method (8) by definition1 is zero stable and by Henrici (1962), the
Therefore, 1
block method is convergent since it is also consistent.
Similarly, the block method (12) expressed in the form of (14) gives

1

 −1
 −1

 −1
 −1


0 0 0 0   yn +1   0
 

1 0 0 0   yn + 2   0
0 1 0 0   yn + 3  =  0
 

0 0 1 0   yn + 4   0


0 0 0 1   yn + 5   0



 1427
1440
 637
 1440
h  34
90
 69
 160
 14
 45

−798
1440
1022
1440
114
90
174
160
64
45

482
1440
−258
1440
34
90
174
160
24
45

−173
1440
77
1440
−1
90
69
160
64
45


−11 
1440 
0 

−3
160 
14  
45  

27
1440

0 0 0 1   yn − 4 


0 0 0 0   yn − 3 
0 0 0 0   yn − 2  +


0 0 0 0   yn −1 


0 0 0 0   yn 

f n +1   0
 
f n+ 2   0
f n +3  +  0
 
f n+ 4   0

f n +5   0


0 0 0
0 0 0
0 0 0
0 0 0
0 0 0




−1  
90

−3
160  

0 

475
1440
−27
1440

f n− 4  

f n −3  
f n− 2  

f n −1  

fn 

(17)

where

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Vol.3, No.11, 2013

A(0)

1

 −1
=  −1

 −1
 −1


B (0)

 1427
1440
 637
 1440
=  34
90
 69
 160
 14
 45

Substituting

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0 0 0 0
0 0


1 0 0 0
0 0
(1)
0 1 0 0, A =  0 0


0 0 1 0
0 0


0 0 0 1
0 0
−798
1440
1022
1440
114
90
174
160
64
45

482
1440
−258
1440
34
90
174
160
24
45

−173
1440
77
1440
−1
90
69
160
64
45

0 0 1

0 0 0
0 0 0

0 0 0

0 0 0


0 0


−11
1440 
0 0
(1)
0 , B = 0 0


−3
160 
0 0
0 0
14 

45 

27
1440

0 0
0 0
0 0
0 0
0 0




−1 
90

−3
160 

0 

475
1440
−27
1440

A(0) and A(1) into (15) gives the characteristic polynomial of the block method (12) as
ρ (λ ) = det(λ A(0) − A(1) )
 1
 
  −1
= det λ  −1
 
  −1
  −1
 
λ
 −λ

= det  −λ

 −λ
 −λ


0 0 0 0 0
 
1 0 0 0 0
0 1 0 0 − 0
 
0 0 1 0 0

0 0 0 1 0


0

0

0

λ

0

0

0 λ 0
0 0 λ
0

0

0

0 0 0 1 

0 0 0 0 
0 0 0 0 

0 0 0 0 

0 0 0 0 


−1
0

0

0
λ


= λ5 − λ4 = 0
λ = 1, λ2 = λ3 = λ4 = λ5 = 0 . The block method (12) by definition1 is zero stable and by Henrici
Therefore, 1
(1962), it is convergent since it is also consistent.
3.2 Regions of Absolute Stability of the Methods
The absolute stability regions of the newly constructed block methods (8) and (12) are plotted using Chollom
(2005) by reformulating the methods as general linear methods. The regions of absolute stability of the methods
of the main discrete schemes and the new block methods are as shown in the figures 1 and 2 below. These
absolute stability regions are all A –stable since they consist of the set of points in the complex plane outside the
enclosed figures.
4 Numerical Examples
We report here a few numerical examples on some stiff problems taken from the literature. For comparisons, we
also report the performance of the new block methods and the well-known Matlab stiff ODE solver ODE23S on
the same problems.

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Vol.3, No.11, 2013

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Problem 1: We consider the Robertson’s problem

y '1 = −0.04 y1 + 10000 y2 y3

y1 (0) = 1

2
y '2 = 0.04 y1 − 10000 y2 y3 − 30000000 y2

y2 (0) = 0

2
y '3 = 30000000 y2

y3 (0) = 0

0 ≤ x ≤ 70, h = 0.1
Problem 2: We consider the Van der Pol’s equation

y '1 = y2

y1 (0) = 2

y '2 = - y1 + µ y2 (1- y12 ) ,

y2 (0) = 0

0 ≤ x ≤ 70, h = 0.01, µ = 10,
5. Conclusion
The new block methods were applied to two well-known stiff problems from the literature, the numerical results
suggest that the proposed new block methods (8) and (12) are suitable for solving stiff problems and perform
competitively with the well-known ODE23s. This success is achieved because of the good stability properties of
the proposed new block methods.
References
Brugnano, L. and D, Trigiante. (1998). Solving Differential Problems by Multistep Initial and Boundary Value
Methods. Gordon and Breach Science Publishers, Amsterdam.
Chollom, J.P., (2005). The construction of Block Hybrid Adam Moulton Methods with link to Two Step RungeKutta Methods. (Ph.D) Thesis University of Jos.
Chollom, J.P., Ndam, J.N. and Kumleng, G.M. (2007). On Some Properties of the Block Linear Multistep
Methods. Science World Journal, 2(3), 11 – 17.
Chollom, J.P., Olatunbasun,I, O. and Omagu, S. (2012). A Class of A-Stable Block Explicit Methods for the
Solutions of ODEs Research Journal of Mathematics and Statistics, 4(2): 52-56.
Fatunla, S.O. (1991). Parallel Method for Second Order Differential Equations. Proceedings of the National
Conference on Computational Mathematics, University of Ibadan Press, pp:87-99.
Fatunla, S.O. (1994). Block Method for Second Order Differential Equations. International Journal of
Computer Mathematics. 41,55 – 63.
Henrici, P. (1962). Discrete Variable Methods in Ordinary Differential Equations. John Wiley, New York, p407
Kim, P. (2010). An Explicit Type Stable Method for Solving Stiff Initial Value Problems. Presentation at a mini
workshop at Knu. Republic of Korea, Retrieved from: http://webbuild.knu.ac.kr/~skim/conf_math2/kimps.pdf.
Kumleng,G.M. and Skwame,Y. (2011) A New A-Stable Method for the Solution of Stiff Initial Value
Problems. International Journal of Numerical Maths.6 (2), 360 -373.
Kumelng,G.M., Sirisena, U.W. and Chollom, J.P.(2012) Construction of a Class of Block Hybrid Implicit
Multistep Methods for the Solution of Stiff Ordinary Differential Equations. Nigerian Journal of Pure and
Applied Sciences 5.
Lie, I. and Norsett, S.P. (1989). Super convergence for Multistep Collocation. Mathematics of Computation.
52(185), 65 – 79.
Onumanyi, P., Awoyemi, D.O., Jator, S.N. and Sirisena, U.W. (1994).New Linear Multistep Methods with
Continuous Coefficients for First Order Initial Value Problems. Journal of the Nigerian Mathematical
Society .13, 37 – 51.

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2.5
2
1.5
1

I m (z )

0.5
0
k=3

k=5

-0.5
-1
-1.5
-2
-2.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Re(z)

Figure2. Absolute Stability Regions of the new Three and Five step discrete methods
2.5
2
1.5
1

I m (z )

0.5
0
k=5

k=3

-0.5
-1
-1.5
-2
-2.5
-0.5

0

0.5

1
Re(z)

1.5

2

2.5

Figure1. Absolute Stability Regions of the Three and Five step New block methods

The absolute stability regions consists of the set of points in the complex plane outside the enclosed figures.
Therefore, both the discrete and block methods are all A - stable since the left –half complex plane is contained
in S., where S = { z ∈ C : R ( z ) ≤ 1} .

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1

y1 Num
y2 Num
y3 Num
y1 ODE23s
y2 ODE23s
y3 ODE23s

0.9
0.8
0.7
S o lu t io n (y )

0.6
0.5
0.4
0.3
0.2
0.1
0

0

10

20

30

40

50

60

70

Time(t)

Figure 1. Solution to Problem 1 using the Three Step Block Method and ODE23S
y1 Num
y2 Num
y3 Num
y1 ODE23s
y2 ODE23s
y3 ODE23s

1
0.9
0.8

S o lu tio n (y )

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0

10

20

30

40

50

60

70

Time(t)

Figure 2. Solution to Problem 1 using the Five Step Block Method and ODE23S

15

y1 Num
y2 Num
y1 ODE23s
y2 ODE23s

10

S o lu t io n ( y )

5

0

-5

-10

-15

0

10

20

30

40

50

60

70

Time(t)

Figure 3. Solution to Problem 2 using the Three Step Block Method and ODE23S

8

Vol.3, No.11, 2013

www.iiste.org

15
y1 Num
y2 Num
y1 ODE23s
y2 ODE23s

10

S o lu t io n ( y )

5

0

-5

-10

-15

0

10

20

30

40

50

60

70

Time(t)

Figure 4. Solution to Problem 2 using the Five Step Block Method and ODE23S
Table 1: Order and Error Constants of the new three step method
Method

y n +1

Order
4

Error constant

−

4

y n+2

1
− 90
3
− 80

4

y n+3

19
720

Table 2: Order and Error Constant of the new five step method
Method

y n +1
y n+2
y n+3

Order
6

Error constant
863
− 60480
271
60480

6

1
756

6

y n+4
y n+5

13
2240

6

6

8
− 945

9

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A class of a stable order four and six linear multistep methods for stiff initial value problems

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A class of a stable order four and six linear multistep methods for stiff initial value problems