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March  2017, 7(1): 95-106. doi: 10.3934/naco.2017006

Adaptive order of block backward differentiation formulas for stiff ODEs

1. 

Institute for Mathematical Research, Department of Mathematics, University Putra Malaysia, 43400 UPM Serdang, Selangor Malaysia

2. 

Department of Mathematics, Pusat Asasi Pertahanan, Universiti Pertahanan Nasional Malaysia, Kem Sungai Besi, 57000 Kuala Lumpur, Malaysia

3. 

Department of Mathematics, Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 UiTM Shah Alam, Selangor Darul Ehsan, Malaysia

4. 

Department of Fundamental and Applied Sciences, Faculty of Science and Information Technology, Universiti Teknologi PETRONAS, 32610 Bandar Seri Iskandar, Perak, Malaysia

* Corresponding author: Z. B. Ibrahim, Email Address: zarinabb@upm.edu.my

Received  June 2016 Revised  August 2016 Published  February 2017

In this paper, Adapative Order of Block Backward Differentiation Formulas (ABBDFs) are formulated using uniform step size for the numerical solution of stiff ordinary differential equations (ODEs). These ABBDF methods are of order four, five and six. The benefit of the ABBDF methods is the computation time in the computation of solutions. Numerical results are presented to demonstrate the advantage of implementing adaptive order selection in a single code.

Citation: Z. B. Ibrahim, N. A. A. Mohd Nasir, K. I. Othman, N. Zainuddin. Adaptive order of block backward differentiation formulas for stiff ODEs. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 95-106. doi: 10.3934/naco.2017006
References:
[1]

L. G. Birta and O. Abou-Rabia, Parallel block predictor corrector methods for ODEs, IEEE Transactions on Computer, 36 (1987), 299-311. Google Scholar

[2]

L. Chu and H. Hamilton, Parallel solution of ODEs by multi-block methods, SIAM J. Sci. Statist. Computation, 8 (1987), 342-353. doi: 10.1137/0908039. Google Scholar

[3]

S. O. Fatunla, Numerical Methods for Initial Value Problems for Ordinary Differential equations 1st edition, U. S. A Academy Press, Boston, 1988. Google Scholar

[4]

P. J. Houwen and B. P. Sommeijer, Block Runge-Kutta Methods on Parallel Computers Report NM-R8906, Centre for Mathematics and Computer Science, Amsterdam, 1989.Google Scholar

[5]

P. J. Houwen and B. P. Sommeijer, Block Runge-Kutta methods on parallel computers Report NM-R8906, Centre for Mathematics and Computer Science, Amsterdam.Google Scholar

[6]

Z. B. IbrahimK. I. Othman and M. B. Suleiman, Implicit r-point block backward differentiation formula for solving first-order stiff ODEs, Applied Mathematics and Computation, 186 (2007), 558-565. doi: 10.1016/j.amc.2006.07.116. Google Scholar

[7]

Z. B. IbrahimK. I. Othman and M. B. Suleiman, Variable step size block backward differentiation formula for solving stiff ODEs, Proc. of World Congress on Engineering, LONDON, U.K, 2 (2007), 785-789. Google Scholar

[8]

Z. B. IbrahimM. B. Suleiman and K. I. Othman, Fixed coefficients block backward differentiation formulas for the numerical solution of stiff ordinary differential equations, European Journal of Scientific Research, 21 (2008), 508-520. Google Scholar

[9]

J. D. Lambert, Numerical Methods for Ordinary Differential Systems 2nd edition, John Willey and Sons, New York, 1991. Google Scholar

[10]

N. E. Mastorakis, An extended Crank-Nicholson method and its Applications in the Solution of Partial Differential Equations: 1-D and 3-D Conduction Equations, 10th WSEAS International Conference on Applied Mathematics, (2006), 134-143. Google Scholar

[11]

Z. A. Majid, Parallel Block Methods for Solving Ordinary Differential Equations Ph. D thesis, Faculty of Science, Universiti Putra Malaysia, 2004.Google Scholar

[12]

N. E. Mastorakis and O. Martin, About the numerical solution of a stationary transport equation, 5th WSEAS Int. Conf. on Simulation, Modeling and Optimization, (2005), 419-426. Google Scholar

[13]

N. A. A. M. NasirZ. B. IbrahimK. I. Othman and M. B. Suleiman, Fifth order two-point block backward differentiation formulas for solving ordinary differential equations, Applied Mathematical Sciences, 5 (2011), 3505-3518. Google Scholar

[14]

L. F. Shampine, Order selection in ODE codes based on implicit formulas, Applied Mathematics Letters, 4 (1991), 53-55. doi: 10.1016/0893-9659(91)90168-U. Google Scholar

[15]

L. F. Shampine and H. A. Watts, Block implicit one-step methods, Math. Comp., 23 (1969), 731-740. Google Scholar

show all references

References:
[1]

L. G. Birta and O. Abou-Rabia, Parallel block predictor corrector methods for ODEs, IEEE Transactions on Computer, 36 (1987), 299-311. Google Scholar

[2]

L. Chu and H. Hamilton, Parallel solution of ODEs by multi-block methods, SIAM J. Sci. Statist. Computation, 8 (1987), 342-353. doi: 10.1137/0908039. Google Scholar

[3]

S. O. Fatunla, Numerical Methods for Initial Value Problems for Ordinary Differential equations 1st edition, U. S. A Academy Press, Boston, 1988. Google Scholar

[4]

P. J. Houwen and B. P. Sommeijer, Block Runge-Kutta Methods on Parallel Computers Report NM-R8906, Centre for Mathematics and Computer Science, Amsterdam, 1989.Google Scholar

[5]

P. J. Houwen and B. P. Sommeijer, Block Runge-Kutta methods on parallel computers Report NM-R8906, Centre for Mathematics and Computer Science, Amsterdam.Google Scholar

[6]

Z. B. IbrahimK. I. Othman and M. B. Suleiman, Implicit r-point block backward differentiation formula for solving first-order stiff ODEs, Applied Mathematics and Computation, 186 (2007), 558-565. doi: 10.1016/j.amc.2006.07.116. Google Scholar

[7]

Z. B. IbrahimK. I. Othman and M. B. Suleiman, Variable step size block backward differentiation formula for solving stiff ODEs, Proc. of World Congress on Engineering, LONDON, U.K, 2 (2007), 785-789. Google Scholar

[8]

Z. B. IbrahimM. B. Suleiman and K. I. Othman, Fixed coefficients block backward differentiation formulas for the numerical solution of stiff ordinary differential equations, European Journal of Scientific Research, 21 (2008), 508-520. Google Scholar

[9]

J. D. Lambert, Numerical Methods for Ordinary Differential Systems 2nd edition, John Willey and Sons, New York, 1991. Google Scholar

[10]

N. E. Mastorakis, An extended Crank-Nicholson method and its Applications in the Solution of Partial Differential Equations: 1-D and 3-D Conduction Equations, 10th WSEAS International Conference on Applied Mathematics, (2006), 134-143. Google Scholar

[11]

Z. A. Majid, Parallel Block Methods for Solving Ordinary Differential Equations Ph. D thesis, Faculty of Science, Universiti Putra Malaysia, 2004.Google Scholar

[12]

N. E. Mastorakis and O. Martin, About the numerical solution of a stationary transport equation, 5th WSEAS Int. Conf. on Simulation, Modeling and Optimization, (2005), 419-426. Google Scholar

[13]

N. A. A. M. NasirZ. B. IbrahimK. I. Othman and M. B. Suleiman, Fifth order two-point block backward differentiation formulas for solving ordinary differential equations, Applied Mathematical Sciences, 5 (2011), 3505-3518. Google Scholar

[14]

L. F. Shampine, Order selection in ODE codes based on implicit formulas, Applied Mathematics Letters, 4 (1991), 53-55. doi: 10.1016/0893-9659(91)90168-U. Google Scholar

[15]

L. F. Shampine and H. A. Watts, Block implicit one-step methods, Math. Comp., 23 (1969), 731-740. Google Scholar

Figure 1.  Stability region of BBDF order 4, 5 and 6
Figure 2.  k-order ABBDF method
Figure 3.  Problem 3.1
Figure 4.  Problem 3.2
Table 1.  Coefficients for ABBDF
$k$$y_{n-4}$$y_{n-3}$$y_{n-2}$$y_{n-1}$$y_{n}$$y_{n+1}$$y_{n+2}$$f_{n+1}$$f_{n+2}$
400$\frac{1}{10}$$-\frac{3}{5}$ $\frac{9}{5}$1 $-\frac{3}{10}$ $\frac{12}{10}$0
00$-\frac{3}{25}$$\frac{16}{25}$ $-\frac{36}{25}$ $\frac{48}{25}$10 $\frac{12}{25}$
50$-\frac{3}{65}$$\frac{4}{13}$$-\frac{12}{13}$ $\frac{24}{13}$1 $\frac{12}{65}$ $\frac{12}{13}$0
0$\frac{12}{137}$$-\frac{75}{137}$$\frac{200}{137}$ $-\frac{300}{137}$ $\frac{300}{137}$10 $\frac{16}{137}$
6$\frac{2}{77}$-$\frac{15}{77}$$\frac{50}{77}$$-\frac{100}{77}$ $\frac{150}{77}$1 $-\frac{10}{77}$ $\frac{60}{77}$0
-$\frac{10}{147}$$\frac{24}{49}$$-\frac{75}{49}$$\frac{400}{147}$ $-\frac{150}{49}$ $\frac{120}{49}$10 $\frac{60}{147}$
$k$$y_{n-4}$$y_{n-3}$$y_{n-2}$$y_{n-1}$$y_{n}$$y_{n+1}$$y_{n+2}$$f_{n+1}$$f_{n+2}$
400$\frac{1}{10}$$-\frac{3}{5}$ $\frac{9}{5}$1 $-\frac{3}{10}$ $\frac{12}{10}$0
00$-\frac{3}{25}$$\frac{16}{25}$ $-\frac{36}{25}$ $\frac{48}{25}$10 $\frac{12}{25}$
50$-\frac{3}{65}$$\frac{4}{13}$$-\frac{12}{13}$ $\frac{24}{13}$1 $\frac{12}{65}$ $\frac{12}{13}$0
0$\frac{12}{137}$$-\frac{75}{137}$$\frac{200}{137}$ $-\frac{300}{137}$ $\frac{300}{137}$10 $\frac{16}{137}$
6$\frac{2}{77}$-$\frac{15}{77}$$\frac{50}{77}$$-\frac{100}{77}$ $\frac{150}{77}$1 $-\frac{10}{77}$ $\frac{60}{77}$0
-$\frac{10}{147}$$\frac{24}{49}$$-\frac{75}{49}$$\frac{400}{147}$ $-\frac{150}{49}$ $\frac{120}{49}$10 $\frac{60}{147}$
Table 2.  Comparison between the ABBDF and BBDF for solving Problem 3.1
$H$MethodMAXETIME
$10^{-2}$BBDF(4)5.34238e-60.008280686
BBDF(5)8.60443e-60.008490485
BBDF(6)1.09196e-50.0089108066
ABBDF6.29854e-60.0082606264
$10^{-4}$BBDF(4)7.07097e-120.0145041159
BBDF(5)3.99341e-110.01477376
BBDF(6)5.75778e-110.01497369
ABBDF2.51638e-110.014267131
$10^{-6}$BBDF(4)1.19693e-120.331891367
BBDF(5)1.27653e-120.35464837
BBDF(6)1.54599e-120.366591181
ABBDF1.32749e-120.3299601339
$10^{-8}$BBDF(4)5.70600e-1132.31383741
BBDF(5)6.27814e-1133.986499
BBDF(6)8.38467e-1135.61432122
ABBDF6.87739e-1132.222682490
$H$MethodMAXETIME
$10^{-2}$BBDF(4)5.34238e-60.008280686
BBDF(5)8.60443e-60.008490485
BBDF(6)1.09196e-50.0089108066
ABBDF6.29854e-60.0082606264
$10^{-4}$BBDF(4)7.07097e-120.0145041159
BBDF(5)3.99341e-110.01477376
BBDF(6)5.75778e-110.01497369
ABBDF2.51638e-110.014267131
$10^{-6}$BBDF(4)1.19693e-120.331891367
BBDF(5)1.27653e-120.35464837
BBDF(6)1.54599e-120.366591181
ABBDF1.32749e-120.3299601339
$10^{-8}$BBDF(4)5.70600e-1132.31383741
BBDF(5)6.27814e-1133.986499
BBDF(6)8.38467e-1135.61432122
ABBDF6.87739e-1132.222682490
Table 3.  Comparison between the ABBDF and BBDF for solving Problem 3.2
$H$MethodMAXETIME
$10^{-2}$BBDF(4)6.13882e+10.008227781
BBDF(5)2.14167e-10.0082600851
BBDF(6)2.96326e+00.00828741
ABBDF1.44159e-20.008064681
$10^{-4}$BBDF(4)7.76160e-80.01543074
BBDF(5)9.48795e-80.016383168
BBDF(6)1.11123e-70.0171090074
ABBDF7.76160e-80.0142214673
$10^{-6}$BBDF(4)1.93315e-112.97284740
BBDF(5)1.07853e-113.12728618
BBDF(6)2.66430e-113.8798845185
ABBDF1.93315e-112.31853289
$10^{-8}$BBDF(4)2.35567e-984.36481221
BBDF(5)4.41047e-1085.76521578
BBDF(6)2.59949e-987.69974205
ABBDF2.35567e-984.36227574
$H$MethodMAXETIME
$10^{-2}$BBDF(4)6.13882e+10.008227781
BBDF(5)2.14167e-10.0082600851
BBDF(6)2.96326e+00.00828741
ABBDF1.44159e-20.008064681
$10^{-4}$BBDF(4)7.76160e-80.01543074
BBDF(5)9.48795e-80.016383168
BBDF(6)1.11123e-70.0171090074
ABBDF7.76160e-80.0142214673
$10^{-6}$BBDF(4)1.93315e-112.97284740
BBDF(5)1.07853e-113.12728618
BBDF(6)2.66430e-113.8798845185
ABBDF1.93315e-112.31853289
$10^{-8}$BBDF(4)2.35567e-984.36481221
BBDF(5)4.41047e-1085.76521578
BBDF(6)2.59949e-987.69974205
ABBDF2.35567e-984.36227574
Table 4.  Comparison between the ABBDF and mathod in [13] for solving Problem 3.3
$H$MethodMAXE
$10^{-3}$BBDF(5)7.43088e-3
BDF4.97642e+43
ode15s6.30000e-3
ABBDF2.02095e-3
$10^{-5}$BBDF(5)7.68218e-4
BDF7.88641e-4
ode15s6.40000e-3
ABBDF6.22269e-4
$10^{-7}$BBDF(5)8.09607e-8
BDF8.39232e-8
ode15s4.30000e-5
ABBDF6.49214e-8
$H$MethodMAXE
$10^{-3}$BBDF(5)7.43088e-3
BDF4.97642e+43
ode15s6.30000e-3
ABBDF2.02095e-3
$10^{-5}$BBDF(5)7.68218e-4
BDF7.88641e-4
ode15s6.40000e-3
ABBDF6.22269e-4
$10^{-7}$BBDF(5)8.09607e-8
BDF8.39232e-8
ode15s4.30000e-5
ABBDF6.49214e-8
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