© Maxwell Scientific Organization, 2013
Submitted: August 31, 2013 Accepted: September 10, 2013 Published: November 30, 2013
Corresponding Author: G.M. Kumleng, Department of Mathematics, University of Jos, P.M.B. 2084, Plateau State, Nigeria
A Modified Block Adam Moulton (MOBAM) Method for the Solution of Stiff Initial Value Problems of Ordinary Differential Equations
G.M. Kumleng, J.P. Chollom and S. Longwap
Department of Mathematics, University of Jos, P.M.B. 2084, Plateau State, Nigeria
Abstract: Stiff ordinary differential equations pose computational difficulties as they present severe step size restrictions on the numerical methods to be used. Construction of numerical methods that possess suitable stability properties for the solution of such systems has been the target of many researchers. Development of methods suitable for these systems of equations has been either through the use of derivative of the solution or by introducing off-step points, additional stages or super future points. These processes have been exploited in Runge-Kutta methods or linear multistep methods. In this study, an improved class of linear multistep block method has been constructed based on Adams Moulton block methods. The improved methods are shown to be A-stable, a property desirable to handle stiff ODEs. Methods of uniform orders 10 and 11 have been constructed. The efficiency of the new methods tested on stiff systems of ODEs and the results reveal that the MOBAM methods compare favourably with results obtained using the state of the art Matlab Ode23 solver.
Keywords: Block method, initial value problems, non-linear, stability INTRODUCTION
Stiff initial value problems of ordinary differential equations were realized in 1952 arising from the study of chemical kinetics. Since then, many researches have been involved in the study and development of suitable numerical methods to handle them. Curtiss and Hirschfelder (1952), Mitchell and Craggs (1953) and Gear (1971) developed the Backward Differential Formulae which were used to solve stiff problems arising from chemical kinetics. Intensive work done in this regard yielding several efficient numerical algorithms has appeared in the literature. Because of the severe restriction on step size placed on Adams Moulton methods for the solution of stiff ODEs, Garfinket et al. (1978) introduced the concept of A- stability for linear multistep methods. This property became the minimum requirement for any linear multistep method to be used for the solution of stiff ODEs. Achieving A-stability was difficult, other stability requirements such as A(α)-stability and stiffly stability were considered. Chakrararti and Kamel (1983) developed stiffly stable second derivative methods with high order and improved stability regions to handle stiff problems. Several other researchers such as Samir (2013), Evelyn and Renante (2006), Song (2010), Vlachos et al. (2009) and Ali and Gholamreza (2012) developed improved and more friendly numerical methods with improved stability properties that could handle stiff ODEs with various degrees of stiffness.
In this study, we pursue the path of Adams Moulton methods by constructing A-stable block linear multistep methods of uniform order 10 and 11. These methods are obtained by modifying the Adams Moulton method of step number 9 by reducing its interpolation step number from 8 to 3. This approach is similar to that of Brugnano and Trigiante (1998) where they developed the Generalized Adams methods.
The modified methods are constructed using the concept of multistep collocation of Lie and Norsett (1989) which Onumanyi et al. (1994, 1999) referred to as block linear multistep methods. The block methods are arrived at by evaluating the continuous formulation of the new method at grid and off grid points to yield the discrete schemes used as block integrators. The methods are applied simultaneously producing self- starting methods thus eliminating the issue of overlap of pieces of solutions.
CONSTRUCTION OF THE NEW METHOD The k-step general linear multistep method for numerically solving the differential equation:
0 0
' ( , ), ( ) , [ , ],
y = f x y y x =y x∈ a b y∈R (1) is described by the linear difference equation:
∑
∑
= += + = k
j
j n j k
j
j n
j x y h x f
0 0
) ( )
( β
α (2)
Res. J. Math. Stat., 5(4): 32-42, 2013
With h being the step size, αj(x)and βj(x) being the continuous coefficients of the method.
The new improved nine step method based on the Adams Moulton method is defined as:
1 1
0
( ) ( )
m
n v v n v j n j
j
y+ α− x y+ − h β x f+
=
− =
∑
(3) where, v= k2−1,αv−1 and βj( )x are the continuous coefficients of the method, m is the number of distinct collocation points, h is the step size and k=m−1=3,5,7,9,...Using the method in Onumanyi et al. (1994, 1999) and further studied by Kumleng et al. (2012) and Chollom et al. (2012) the method (3) for
v =
k2−1 and j =0, 1… 9 sequence in the matrix form:
I
DC= (4) where, D=C−1, C−1 being the elements of the continuous coefficients of the method.
Construction of MOBAM K = 9: This method has its general form as:
9 ,..., 1 , 0 , ) ( )
( ) (
0 3
3 + =
=
∑
= +
+ h x f j
y x x y
k
j
j n j
n β
α (5)
Using the procedure in Onumanyi et al. (1999), we obtain the D matrix for (5) as:
=
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + + +
9 9 8
9 7
9 6
9 5
9 4
9 3
9 2
9 9
9 8 8
8 7
8 6
8 5
8 4
8 3
8 2
8 8
9 7 8
7 7
7 6
7 5
7 4
7 3
7 2
7 7
9 6 8
6 7
6 6
6 5
6 4
6 3
6 2
6 6
9 5 8
5 7
5 6
5 5
5 4
5 3
5 2
5 5
9 4 8
4 7
4 6
4 5
4 4
4 3
4 2
4 4
9 3 8
3 7
3 6
3 5
3 4
3 3
3 2
3 3
9 2 8
2 7
2 6
2 5
2 4
2 3
2 2
2 2
9 1 8
1 7
1 6
1 5
1 4
1 3
1 2
1 1
9 8 7 6 5 4 3 2
10 3 9
3 8
3 7
3 6
3 5
3 4
3 3
3 2
3 3
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0 1
n n n n n n n n n
n n n n n n n n n
n n n n n n n n n
n n n n n n n n n
n n n n n n n n n
n n n n n n n n n
n n n n n n n n n
n n n n n n n n n
n n n n n n n n n
n n n n n n n n n
n n n n n n n n n n
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x x
D
(6)
Using the Maple software, the inverse of the matrix (6) is obtained and represented as C, the elements of which yield the continuous coefficients α3(x) and βj(x), j = 0,1,…9 of the method (5) as:
) (
) (
) (
) (
) (
) (
) (
) (
) (
) (
) (
) (
) (
) (
) (
) (
) (
) (
) (
) (
1 ) (
9 10 8 9 7 8 6 7 5 6 4 5 3 4 2 3 2
9 10 8 9 7 8 6 7 5 6 4 5 3 4 2 3 2
9 10 8 9 7 8 6 7 5 6 4 5 3 4 2 3 2
9 10 8 9 7 8 6 7 5 6 4
5 3
4 2
3 2
9 10 8 9 7 8 6 7 5 6 4
5 3
4 2
3 2
9 10 8 9 7 8 6 7 5 6 4
5 3
4 2
3 2
9 10 8 9 7 8 6 7 5
6 4
5 3
4 2
3 2
9 10 8 9 7 8 6 7 5 6 4
5 3
4 2
3 2
9 10 8 9 7 8 6 7 5 6 4 5 3
4 2
3 2
9 10 8 9 7 8 6 7 5 6 4 5 3 4 2 3 2
3628800 90720
69120 13 560 103680
1069 2400
89 362880 29531 7560
761 18 3584
25 9
403200 362880
37 23040
41 20160
347 34560
3487 28800 10579 2240 1823 33360
3407 16 9 12800
909 8
100800 45360
19 20160
151 5040
373 8640 3817 1800 2939 1120 4101 210 967 7 18 11200 3663 7
43200 12960
13 2880
53 1680 313 8640 9823 2400 10279 8640 84307 540 6709 7 5600 5039 6
28800 648 2304
67 1008
305 17280 32773 720 5273 64 1091 12 265 5 63 44800 73809 5
28800 25920
41 11520
353 10080
3313 17280 36769 14400 122249 320
6519 240 6499 4 63 8960 19107 4
43200 6480
7 1440
31 1680
401 8640 13873 600 4013 4320 72569 270 6289 14 5600 13273 3
100800 90720
43 720
7 5040 563 8640 6787 7200 24901 2240 20837 420 5869 9 1600 279 2
403200 90720
11 23040
59 360
11 34560 7807 7200 7667 4480 14139 840 4609 2 9 89600 147429 1
3628800 72576
96768 29 1344
5 103680
3013 128
19 9072 4523 6048 6515 5040 7129 12800
3591 0
3
h h
h h h h h h h
h
h h h h h h h h h
h
h h h h h h h h h
h
h h h h h h h h h
h
h h h h h h h h h
h
h h h h h h h h h
h
h h h h h h h h h
h
h h h h h h h h h
h
h h h h h h h h h
h
h h
h h h h h h h
h
x x x x x x x x x x
x
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
β β β β β β β β β
µ β
α
+
− +
− +
− +
− +
−
=
− +
− +
− +
− +
−
=
+
− +
− +
− +
− +
−
=
− +
− +
− +
− +
−
=
+
− +
− +
− +
− +
−
=
− +
− +
− +
− +
−
=
+
− +
− +
− +
− +
−
=
− +
− +
− +
− +
−
=
+
− +
− +
− +
− +
−
=
− +
− +
− +
− +
− +
−
=
=
(7)
Res. J. Math. Stat., 5(4): 32-42, 2013
Substituting the continuous coefficients (7) into (5) yields the continuous form of the MOBAM method (8):
3628800 9 90720
69120 13 560 103680
1069 2400
89 362880 29531 7560
761 18 3584
25
403200 8 362880
37 23040
41 20160
347 34560
3487 28800 10579 2240 1823 33360
3407 16 9 12800
909
100800 7 45360
19 20160
151 5040
373 8640 3817 1800 2939 1120 4101 210 967 7 18 11200 3663
43200 6 12960
13 2880
53 1680
313 8640 9823 2400 10279 8640 84307 540
6709 7 5600 5039
28800 5 648 2304
67 1008
305 17280 32773 720
5273 64 1091 12 265 5 63 44800 73809
28800 4 25920
41 11520
353 10080
3313 17280 36769 14400
122249 320
6519 240 6499 4 63 8960 19107
43200 3 6480
7 1440
31 1680
401 8640 13873 600
4013 4320 72569 270
6289 14 5600 13273
100800 2 90720
43 720
7 5040 563 8640 6787 7200 24901 2240
20837 420
5869 9 1600 279
403200 1 90720
11 23040
59 360 11 34560 7807 7200 7667 4480 14139 840
4609 2 9 89600 147429
3628800 72576
96768 29 1344
5 103680
3013 128
19 9072 4523 6048 6515 5040 7129 12800
3591 3
) (
) (
) (
) (
) (
) (
) (
) (
) (
) (
) (
9 10 8 9 7 8 6 7 5 6 4 5 3 4 2
3 2
9 10 8 9 7 8 6 7 5 6 4 5 3 4 2 3 2
9 10 8 9 7 8 6 7 5 6 4 5 3 4 2 3 2
9 10 8 9 7 8 6 7 5 6 4
5 3
4 2
3 2
9 10 8 9 7 8 6 7 5 6 4
5 3
4 2
3 2
9 10 8 9 7 8 6 7 5 6 4
5 3
4 2
3 2
9 10 8 9 7 8 6 7 5
6 4
5 3
4 2
3 2
9 10 8 9 7 8 6 7 5 6 4
5 3
4 2
3 2
9 10 8 9 7 8 6 7 5 6 4 5 3
4 2
3 2
9 10 8 9 7 8 6 7 5 6 4 5 3 4 2 3 2
+ + + +
+ + + +
+ +
+
− +
− +
− +
− +
−
+
− +
− +
− +
− +
−
+ +
− +
− +
− +
− +
−
+
− +
− +
− +
− +
−
+ +
− +
− +
− +
− +
−
+
− +
− +
− +
− +
−
+ +
− +
− +
− +
− +
−
+
− +
− +
− +
− +
−
+ +
− +
− +
− +
− +
−
+
− +
− +
− +
− +
− +
− +
= +
h n h
h h h h
h h h
h
h n h
h h h h h h h
h
h n h
h h h h h h h
h
h n h h h h h h h h
h
h n h h h h h
h h h
h
h n h h h h h
h h h
h
h n h h h h h h h h
h
h n h
h h h h h h h
h
h n h
h h h h h h h
h
h n h
h h h h
h h h
h n
n
f f f f
f f f f
f f y
x y
µ µ
µ µ µ µ
µ µ
µ
µ µ µ
µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ
µ µ
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
µ µ
µ µ µ µ µ µ
µ µ
µ
(8)
where,µ =x−xn andµ∈
[
0,9h]
. Evaluating the continuous scheme (8) at the following points hh h h h h h
h,2 ,4 ,5 ,6 ,7 ,8 ,9 ,
=0
µ yields the MOBAM method (9) for k = 9 used in block for the solution of ODEs:
) 392 2313
306 3980
1494 3798
598 396
90 9 (
) 2025 101635
387400 185200
381550
247150 171760
25400 4675
425 (
) 13 224 5494
17632 10870
17632 5494
224 13
(
) 49 603
3960 42352
95454 95454
42352 3960
603 49 (
) 23 334
2804
46378 139030
46378 2804
334 23 (
) 2497 27467
141304 462320
1166146
4763582 3609968
342136 50315
3969 (
) 3969 42187
206072 617584
1295810
2166334 5597072
3133688 163531
10625 (
) 104 1063
4910 13492
24266
28234 18742
156340 38938
729 (
) 625 6363
29304 80624
147618
191070 212368
15624 147429
25137 (
9 8
7 6
5 4
3 2
1 1400
3 9
9 8
7 6
5
4 3
2 1
290304 3 8
9 8 7
6 5
4 3
2 1
14175 3 7
9 8
7 6
5 4
3 2
1 89600
3 6
8 7
6
5 4
3 2
1 113400
3 5
9 8
7 6
5
4 3
2 1
7257600 3 4
9 8
7 6
5
4 3
2 1
7257600 3 2
9 8
7 6
5
4 3
2 1
113400 3 1
9 8
7 6
5
4 3
2 1
89600 3
+ +
+ +
+ +
+ +
+ +
+
+ +
+ +
+
+ +
+ +
+ +
+ + +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+
+ +
+
+ +
+ +
+ +
+
+ +
+ +
+
+ +
+ +
+ +
+ +
+ +
+
+ +
+ +
+ +
+ +
+ +
+
+ +
+ +
+ +
+ +
+ +
+
+ +
+ +
+
+ +
−
+
− +
− +
−
=
−
− +
+ +
+ +
+
− +
−
=
−
+
−
+ +
+ +
+
−
=
−
− +
−
+ +
+ +
− +
−
=
−
− +
− +
+ +
− +
−
=
−
− +
− +
− +
+
− +
−
=
−
− +
− +
−
+
−
− +
−
=
−
+
− +
−
+
−
−
−
−
=
−
+
− +
−
+
− +
− +
=
−
n n
n n
n n
n n
n n h n n
n n
n n
n
n n
n n
n h
n n
n n n
n n
n n
n n
h n n
n n
n n
n n
n n
n n h
n n
n n
n
n n
n n
n n h
n n
n n
n n
n
n n
n n
n h
n n
n n
n n
n
n n
n n
n h
n n
n n
n n
n
n n
n n
n h
n n
n n
n n
n
n n
n n
n h
n n
f f
f f
f f
f f
f f y
y
f f
f f
f
f f
f f
f y
y
f f f
f f
f f
f f
y y
f f
f f
f f
f f
f f y
y
f f
f
f f
f f
f f y
y
f f
f f
f
f f
f f
f y
y
f f
f f
f
f f
f f
f y
y
f f
f f
f
f f
f f
f y
y
f f
f f
f
f f
f f
f y
y
(9)
Construction of Hybrid MOBAM K = 9, µ=172 : This is the hybrid form of the MOBAM family for k=9. An off step point is inserted in (5) to give its general form as:
2 17 0
3
3( ) ( ) ( ) , 0,1,...,9,
)
( = + + + = =
= +
+
∑
β β µα x y h x f h x f j
x
y u nu
k
j
j n j
n (10)
Following a similar procedure as in above section gives the matrix:
