Research Article
Stability region of two-point variable step–block backward differentiation formulae
Journal of Algorithms & Computational Technology 2017, Vol. 11(2) 192–198 ! The Author(s) 2016 DOI: 10.1177/1748301816680508 journals.sagepub.com/home/act
SAM Yatim1, AI Asnor2 and ZB Ibrahim3
Abstract Stability region of two-point variable step–block backward differentiation formulae method is analysed in this article where we took into consideration on the increment of step size to a factor 1.8. The stability graph for three distinct step size ratios are plotted using Maple software and presented in different graphs. The stability properties are also become part of the analysis that have been studied in this article.
Keywords Backward differentiation formulae, block backward differentiation formulae, stiff ordinary differential equations, stability region Date received: 13 October 2015; accepted: 9 May 2016
Introduction Apparently, the study on numerical methods for the solution of initial value problem (IVP) of Ordinary Differential Equations (ODEs) especially stiff problem has become a famous topic and advanced in study as many researchers have produced new findings in this particular research areas. The existing numerical methods used for solving ODE are designed to find the approximations to the solutions of ODE where it provides an alternative way to some complex problems in real-world nowadays. Apart from that, the numerical methods are classified as single-step methods or multistep methods. In addition, the methods can be divided into explicit methods or implicit methods. Since numerical method has stability limitation on the step size, there are only few methods that can solve stiff problems.1 The ideas of solving first-order stiff problem using block backward differentiation formulae (BBDF) method has been introduced by Ibrahim et al.2 Currently, BBDF method becomes frequently used for solving stiff ODEs. Furthermore, there are variety of solver, which are based on BBDF method that are available to solve stiff ODE developed in literature.3–10 Basically, for method to be of practical importance, it must have a region of absolute stability
to ensure that the method will be able to solve at least for the mildly stiff problems.11 In this article, we are interested to analyse the stability of VS-BBDF method with constant step size, half the step size and increment the step size by a factor 1.8. Besides, the variable step–block backward differentiation formula (VS-BBDF) method with increment of step size to a factor 1.6 and 1.9 have been studied by Ibrahim et al.2 and Yatim et al.7 The formulae for VS-BBDF method of variable step size approach with increment of the step size to a factor 1.8 is as follows (Table 1).
1 School of Distance Education, Universiti Sains Malaysia, Pulau Pinang, Malaysia 2 School of Mathematical Sciences, Universiti Sains Malaysia, Pulau Pinang, Malaysia 3 Department of Mathematics Faculty of Science, Universiti Putra Malaysia, Selangor, Malaysia
Corresponding author: SAM Yatim, School of Distance Education, Universiti Sains Malaysia, 11800 USM, Pulau Pinang, Malaysia. Email:
[email protected]
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution-NonCommercial 3.0 License (http://www. creativecommons.org/licenses/by-nc/3.0/) which permits non-commercial use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage).
Yatim et al.
193 Table 1. The formulae for two-point VS-BBDF method. Step size ratio
Computed points
r¼1
ynþ1
Coefficients of the points 6 3 1 3 9 hfnþ1 ynþ2 þ yn2 yn1 þ yn 5 10 10 5 5 12 48 3 16 hfnþ2 þ ynþ1 yn2 þ 25 yn1 36 25 y n 25 25 25 15 75 3 25 225 hfnþ1 ynþ2 þ yn2 yn1 þ yn 8 128 128 128 128 12 192 2 3 18 hfnþ2 þ ynþ1 yn2 þ yn1 yn 23 115 115 23 23 266 2527 189 9747 17689 hfnþ1 ynþ2 þ yn2 yn1 þ yn 297 13662 550 6325 7425 644 59248 128547 27216 103684 hfnþ2 þ ynþ1 yn2 þ yn1 yn 1425 27075 225625 11875 35625
ynþ2 r¼2
ynþ1 ynþ2
r ¼ 5/9
ynþ1 ynþ2
The general form of the formulae is
where
1 0 ynþ1 0 1 ynþ2 0 1 ynþ1 1 ¼ þh 0 2 0 ynþ2 1 t1 yn1 0 þ þ 2 t2 yn 0
1
and
2
0 1
or the general form of VS-BBDF method in matrixvector as shown in (1).
2
2
fnþ1
fnþ2
ð1Þ
yn3 yn2
Definition 2.2. The LMM (2) is said to be A-stable if its region of absolute stability contains the whole of the lefthand half-plane ReðhlÞ 5 0.
are the value of previous points.
Stability and its properties Our goal is to determine the stability properties for VSBBDF method when the step size is constant, halved and increased to a factor 1.8. There are some definition on stability of linear multistep method (LMM) from Lambert12 that will be defined later in this section. The general LMM is k X
j ynþj ¼ h
j¼0
k X
j fnþj
ð2Þ
Aj Ynþj ¼ h
j¼0
k X
j¼0
ij ynþ2j ¼ h
Stability analysis of VS-BBDF method Apply (1) to the test equation, f ¼ y0 ¼ ly and then obtain
Bj Fnþj
1 0
0 1
ynþ1 ynþ2
ð3Þ
0 1 ynþ1 1 þh 2 0 ynþ2 0 1 t1 yn1 0 þ þ 2 t2 yn 0
¼
lynþ1 lynþ2 yn3 1 yn2 2
0 2
ð5Þ
j¼0
where Aj and Bj are r by r matrices with elements im , im for i m ¼ 0, 1, . . . , r. Then, apply (3) to VS-BBDF method and yield the general form of VS-BBDF method as below k X
Definition 2.3. The LMM (2) is said to be zero stable of no root of the first characteristic polynomial pðrÞ has modulus greater than one, and if every root with unit modulus is simple.
j¼0
Therefore, the general form of a block LMM is k X
Definition 2.1. The LMM (2) is said to be absolute stable in a region R for a given hl, all the roots rs of the stability polynomial ðr; hlÞ :¼ ðrÞ hlðrÞ ¼ 0, satisfy jrs j 5 1, where s ¼ 1, 2, . . . , k.
k X j¼0
ij fnþ2j
ð4Þ
or let H ¼ hl and yield
1 1 H
1
2
1 2 H
ynþ1 ynþ2
¼
1
t1
yn1
2 t2 yn 0 yn3 1 þ 0 yn2 2 ð6Þ
194
Journal of Algorithms & Computational Technology 11(2)
In simpler way,
Stability polynomial of all step size ratios are as follows,
A0 Yj ¼ A1 Yj1 þ A2 Yj2
ð7Þ . r ¼ 1,
where n ¼ 2j. The stability polynomial of the method is Rðt; HÞ ¼ det
k X
Rðt; HÞ ¼
! Aj t
kj
¼0
ð8Þ
j¼0
Hence (8) is equivalent to
. r ¼ 2,
Rðt; HÞ ¼ detA0 t 2 A1 t A2 ¼ 0
ð9Þ
Rðt; HÞ ¼
For r ¼ 1, 2
3 2 3 6 3 1 H 6 7 6 5 5 10 A0 ¼ 4 5, A1 ¼ 4 48 12 16 1 H 25 25 25 2 1 3 0 6 10 7 A2 ¼ 4 5 3 0 25
9 3 5 7, 5 36 25
3 15 75 H 6 8 128 7, A0 ¼ 4 5 192 12 1 H 115 23 2 3 3 0 6 280 7 A2 ¼ 4 5 2 0 115 1
2
25 225 3 6 7 A1 ¼ 4 128 128 5, 3 18 23 23
For r ¼ 5/9, 2
3 266 2527 H 6 297 13662 7, A0 ¼ 4 5 59248 644 1 H 27075 1425 2 9747 17689 3 6 7425 7, A1 ¼ 4 6325 5 27216 103684 11875 35625 2 189 3 0 6 550 7 A2 ¼ 4 5 128547 0 225625
91 4 173 3 289 2 1 441 4 t t t þ t t H 46 93 2944 2944 184 45 1155 3 3 t H t 2H þ t 4 H2 46 736 92 ð11Þ
. r ¼ 5/9, Rðt; HÞ ¼
For r ¼ 2, 2
197 4 153 3 9 1 42 t t t2 þ t t 4H 125 125 25 125 25 72 4 2 252 3 18 2 t H t H t H þ 125 125 125 ð10Þ
31291 4 3575389 3 755829 2 t t t 22275 10580625 653125 59049 190106 4 9016 4 2 t t Hþ t H þ 653125 141075 22275 1839404 3 66654 2 t H t H 556875 130625 ð12Þ
The absolute stability region for each step size ratios in the hl plane is investigated by evaluating the stability polynomial. The stability region is the region enclosed by the set of points determined by replacing t ¼ ei ¼ sin þ i cos , 0 5 5 2 in stability polynomial. By the definition mentioned earlier, the stability region is determined by finding the region for which jtj 5 1. Furthermore, the region of stability of VSBBDF method is plotted using MAPLE software. To obtain zero stability, let H ¼ hl ¼ 0 and replace H into all stability polynomials in (10), (11), and (12). Thus, we yield
1
. r ¼ 1,
Rðt; 0Þ ¼
197 4 153 3 9 1 t t t2 þ t 125 125 25 125
ð13Þ
. r ¼ 2,
Rðt; 0Þ ¼
91 4 173 3 289 2 1 t t t þ t 46 93 2944 2944
ð14Þ
Yatim et al.
195 Table 2. Lists of stability polynomials and roots for the three step size ratios. Step size ratio, r
Stability polynomial, Rðt; HÞ 197 4 153 3 9 1 42 t t t2 þ t t4 H 125 125 25 125 25 72 4 2 252 3 18 2 þ t H t H t H 125 125 125 91 4 173 3 289 2 1 441 4 t t t þ t t H 46 93 2944 2944 184 45 1155 3 3 t H t2 H þ t4 H2 46 736 92 31291 4 3575389 3 755829 2 59049 t t t þ t 22275 10580625 653125 653125 190106 4 9016 4 2 1839404 3 66654 2 t Hþ t H t H t H 141075 22275 556875 130625
r¼1
r¼2
r ¼ 5/9
. r ¼ 5/9,
Rðt; 0Þ ¼
31291 4 3575389 3 755829 2 59049 t t t þ t 22275 10580625 653125 653125 ð15Þ
Solve the updated equations will yield the roots for the stability polynomial. Thus, the stability polynomial and roots for all distinct step size ratios of VS-BBDF method are listed in Table 2.
Numerical examples The test problems of stiff ODEs are solved using the VS-BBDF method and we compared the numerical results with ode15s and ode23s in term of maximum errors and number of total steps. The notations are used in the tables and figures. VS-BBDF
TSs
TOL MAXE
Roots
Variable step–block backward differentiation formulae method Total steps taken during the computation of approximate solution Tolerance limit Maximum error
Test Problem 1: y0 ¼ 1000y þ 3000 2000ex Interval: ½0, 20 Exact solution: yðxÞ ¼ 3 0:998e1000x 2:002ex Initial conditions: yð0Þ ¼ 0 Eigenvalue: l ¼ 100 Source: Voss and Abbas13
1, 0.0207917599, 0.2441420137 1, 0.00325762197, 0.05270817143 1, 0.0769489074 0.8363962010
Test Problem 2: y01 ðxÞ ¼ 998y1 þ 1998y2 y02 ðxÞ ¼ 999y1 1999y2 Interval: ½0, 20 y1 ðxÞ ¼ 2ex e1000x Exact solution: y2 ðxÞ ¼ ex þ e1000x Initial conditions: y1 ð0Þ ¼ 1, y2 ð0Þ ¼ 0 Eigenvalues: l ¼ 1, 1000 Source: Gear4 Test Problem 3: y01 ¼ 1195y1 1995y2 y02 ¼ 1197y1 1997y2 Interval: ½0, 20 y1 ðxÞ ¼ 10e2x 8e800x Exact solution: y2 ðxÞ ¼ 6e2x 8e800x Initial conditions: y1 ð0Þ ¼ 2, y2 ð0Þ ¼ 2 Eigenvalues: l ¼ 2, 800 Source: Gerald and Wheatley14
Results for Test Problem 1 Table 3. Numerical results for Test Problem 1. Problem
Method
TOL
‘1
VS-BBDF
102
29
104
56
1.5807e-006
106
135
9.9454e-007
102
40
8.4000e-003
104
93
1.6634e-004
106
165
3.0953e-006
102 104
36 181
4.5000e-003 2.5500e-004
106
1193
1.0911e-005
ode15s
ode23s
TSs
MAXE 1.1090e-004
196
Journal of Algorithms & Computational Technology 11(2)
Results for Test Problem 2
Table 4. Numerical results for Test Problem 2. Problem 2
Method VS-BBDF
ode15s
ode23s
TOL
TSs
MAXE
2
30 61 152 37 89 167 22 67 287
1.1291e-004 1.0537e-007 1.1345e-008 1.7600e-002 1.8659e-004 3.9569e-006 7.3100e-003 3.6837e-004 1.7039e-005
10 104 106 102 104 106 102 104 106
Results for Test Problem 3 Figure 2. Stability region of VS-BBDF method when r ¼ 2. Table 5. Numerical results for Test Problem 3. Problem
Method
TOL
TSs
3
VS-BBDF
102
36
104
76
9.2833e-007
106
196
1.0238e-008
102
52
1.0940e-001
104 106
67 85
1.0940e-001 1.0940e-001
102
30
4.4400e-002
104
37
4.4400e-002
106
43
4.4400e-002
ode15s
ode23s
MAXE 1.1792e-004
Figure 3. Stability region of VS-BBDF method when r ¼ 5/9.
-4
log|Maximum error|
-5 -6 -7 -8 -9 -10 -11 -12 -13 -14
TOL VS-BBDF
Figure 1. Stability region of VS-BBDF method when r ¼ 1.
ode15s
ode23s
Figure 4. Efficiency curves VS-BBDF and Matlab’s ODE solvers for Test Problem 1.
Yatim et al.
197
VS-BBDF
ode15s
ode23s
Total steps
1020 820 620 420 220 20 10e-2
10e-4
10e-6
TOL
-4
-2
-6
-4
log|Maximum Error|
log|Maximum Error|
Figure 5. Total steps curves of VS-BBDF and Matlab’s ODE solvers for Test Problem 1.
-8
-10 -12 -14 -16
-6 -8 -10 -12 -14 -16 -18
-18
-20
-20
TOL
TOL VS-BBDF
ode15s
ode23s
Figure 6. Efficiency curves VS-BBDF and Matlab’s ODE solvers for Test Problem 2.
ode15s
VS-BBDF
ode23s
270
200 180
220
160
170 120 70
ode15s
ode23s
Figure 8. Efficiency curves VS-BBDF and Matlab’s ODE solvers for Test Problem 3.
Total steps
Total steps
VS-BBDF
VS-BBDF
ode15s
ode23s
140 120 100 80 60 40
20 10e-2
10e-4
10e-6
TOL
Figure 7. Total steps curves of VS-BBDF and Matlab’s ODE solvers for Test Problem 2.
Conclusions From Figures 1–3, the stability regions for all the step size ratios lie outside the closed region. Since all the roots for the step size ratios have modulus less than
20 10e-2
10e-4
10e-6
TOL
Figure 9. Total steps curves of VS-BBDF and Matlab’s ODE solvers for Test Problem 3.
or equal to 1, thus the proposed method satisfy the zero stability condition. The figures also show that the absolute stability region for step size ratios 1 and 5/9 are almost A-stable while the stability region for
198 step size ratio 2 is A-stable since the absolute stability region covers the entire left half-plane of the complex plane, ReðhlÞ 5 0. Tables 3–5 present the numerical results for the solver in terms of maximum errors. From Figures 4 to 9, it is proven that the maximum error obtained are within the tolerance given and VS-BBDF method converges faster as compared with ode15s and ode23s. Therefore, we can conclude that the proposed method is suitable for solving the stiff ODE problems. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This paper is supported by Universiti Sains Malaysia under Short Term Grant USM, 304/PJJAUH/6313177.
References 1. Fatunla SO. Block methods for second order ODEs. Intern J Comput Math 1990; 40: 55–63. 2. Ibrahim ZB, Othman KI and Suleiman M. Variable step block backward differentiation formula for solving first order stiff ODEs. In: Proceedings of the world congress on engineering 2007, London, 2007. 3. Abasi N, Suleiman M, Abbasi N, et al. 2-Point block BDF method with off-step points for solving stiff ODEs. J Soft Comput Appl 2014; 2014: 1–15. 4. Gear CW. Numerical initial value problems in ordinary differential equations. New Jersey: Prentice Hall, 1971.
Journal of Algorithms & Computational Technology 11(2) 5. Suleiman MB, Musa H, Ismail F, et al. A new variable step size block backward differentiation formula for solving stiff initial value problems. Int J Comput Math 2013; 90: 2391–2408. 6. Yatim SAM. Variable step variable order block backward differentiation formulae for solving stiff ordinary differential equations. PhD Thesis, Universiti Putra Malaysia, 2013. 7. Yatim SAM, Ibrahim ZB, Othman KI, et al. Fifth order variable step block backward differentiation formula for solving stiff ODEs. World Acad Sci Eng Technol 2010; 4: 236–238. 8. Zawawi ISM, Ibrahim ZB and Othman KI. Derivation of diagonally implicit block backward differentiation formulas for solving stiff initial value problems. Math Prob Eng 2015; 2015: 1–13. 9. Musa H, Suleiman MB and Senu N. Fully implicit 3-point block extended backward differentiation formula for stiff initial value problems. Appl Math Sci 2012; 6: 4211–4228. 10. Yatim SAM, Ibrahim ZB, Othman KI, et al. A numerical algorithm for solving stiff ordinary differential equations. Math Prob Eng 2013; 2013: 1–11. 11. Majid ZA and Suleiman M. Performance of 4-point diagonally method for solving ordinary differential equations. Matematika 2006; 22: 137–146. 12. Lambert JD. Computational methods in ordinary differential equations. New York: John Wiley & Sons, Inc, 1991. 13. Voss D and Abbas S. Block predictor-corrector schemes for the parallel solution of ODEs. Comput Math Appl 1997; 33: 65–72. 14. Gerald CF and Wheatley PO. Applied numerical analysis, 4th ed. Boston: Addison-Wesley Publishing Company, 1989.
