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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 and 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. [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. [3] S. O. Fatunla, Numerical Methods for Initial Value Problems for Ordinary Differential equations 1st edition, U. S. A Academy Press, Boston, 1988. [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. [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. [6] Z. B. Ibrahim, K. 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. [7] Z. B. Ibrahim, K. 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. [8] Z. B. Ibrahim, M. 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. [9] J. D. Lambert, Numerical Methods for Ordinary Differential Systems 2nd edition, John Willey and Sons, New York, 1991. [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. [11] Z. A. Majid, Parallel Block Methods for Solving Ordinary Differential Equations Ph. D thesis, Faculty of Science, Universiti Putra Malaysia, 2004. [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. [13] N. A. A. M. Nasir, Z. B. Ibrahim, K. 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. [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. [15] L. F. Shampine and H. A. Watts, Block implicit one-step methods, Math. Comp., 23 (1969), 731-740.

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##### References:
 [1] L. G. Birta and O. Abou-Rabia, Parallel block predictor corrector methods for ODEs, IEEE Transactions on Computer, 36 (1987), 299-311. [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. [3] S. O. Fatunla, Numerical Methods for Initial Value Problems for Ordinary Differential equations 1st edition, U. S. A Academy Press, Boston, 1988. [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. [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. [6] Z. B. Ibrahim, K. 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. [7] Z. B. Ibrahim, K. 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. [8] Z. B. Ibrahim, M. 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. [9] J. D. Lambert, Numerical Methods for Ordinary Differential Systems 2nd edition, John Willey and Sons, New York, 1991. [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. [11] Z. A. Majid, Parallel Block Methods for Solving Ordinary Differential Equations Ph. D thesis, Faculty of Science, Universiti Putra Malaysia, 2004. [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. [13] N. A. A. M. Nasir, Z. B. Ibrahim, K. 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. [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. [15] L. F. Shampine and H. A. Watts, Block implicit one-step methods, Math. Comp., 23 (1969), 731-740.
Stability region of BBDF order 4, 5 and 6
k-order ABBDF method
Problem 3.1
Problem 3.2
Coefficients for ABBDF
 $k$ $y_{n-4}$ $y_{n-3}$ $y_{n-2}$ $y_{n-1}$ $y_{n}$ $y_{n+1}$ $y_{n+2}$ $f_{n+1}$ $f_{n+2}$ 4 0 0 $\frac{1}{10}$ $-\frac{3}{5}$ $\frac{9}{5}$ 1 $-\frac{3}{10}$ $\frac{12}{10}$ 0 0 0 $-\frac{3}{25}$ $\frac{16}{25}$ $-\frac{36}{25}$ $\frac{48}{25}$ 1 0 $\frac{12}{25}$ 5 0 $-\frac{3}{65}$ $\frac{4}{13}$ $-\frac{12}{13}$ $\frac{24}{13}$ 1 $\frac{12}{65}$ $\frac{12}{13}$ 0 0 $\frac{12}{137}$ $-\frac{75}{137}$ $\frac{200}{137}$ $-\frac{300}{137}$ $\frac{300}{137}$ 1 0 $\frac{16}{137}$ 6 $\frac{2}{77}$ -$\frac{15}{77}$ $\frac{50}{77}$ $-\frac{100}{77}$ $\frac{150}{77}$ 1 $-\frac{10}{77}$ $\frac{60}{77}$ 0 -$\frac{10}{147}$ $\frac{24}{49}$ $-\frac{75}{49}$ $\frac{400}{147}$ $-\frac{150}{49}$ $\frac{120}{49}$ 1 0 $\frac{60}{147}$
 $k$ $y_{n-4}$ $y_{n-3}$ $y_{n-2}$ $y_{n-1}$ $y_{n}$ $y_{n+1}$ $y_{n+2}$ $f_{n+1}$ $f_{n+2}$ 4 0 0 $\frac{1}{10}$ $-\frac{3}{5}$ $\frac{9}{5}$ 1 $-\frac{3}{10}$ $\frac{12}{10}$ 0 0 0 $-\frac{3}{25}$ $\frac{16}{25}$ $-\frac{36}{25}$ $\frac{48}{25}$ 1 0 $\frac{12}{25}$ 5 0 $-\frac{3}{65}$ $\frac{4}{13}$ $-\frac{12}{13}$ $\frac{24}{13}$ 1 $\frac{12}{65}$ $\frac{12}{13}$ 0 0 $\frac{12}{137}$ $-\frac{75}{137}$ $\frac{200}{137}$ $-\frac{300}{137}$ $\frac{300}{137}$ 1 0 $\frac{16}{137}$ 6 $\frac{2}{77}$ -$\frac{15}{77}$ $\frac{50}{77}$ $-\frac{100}{77}$ $\frac{150}{77}$ 1 $-\frac{10}{77}$ $\frac{60}{77}$ 0 -$\frac{10}{147}$ $\frac{24}{49}$ $-\frac{75}{49}$ $\frac{400}{147}$ $-\frac{150}{49}$ $\frac{120}{49}$ 1 0 $\frac{60}{147}$
Comparison between the ABBDF and BBDF for solving Problem 3.1
 $H$ Method MAXE TIME $10^{-2}$ BBDF(4) 5.34238e-6 0.008280686 BBDF(5) 8.60443e-6 0.008490485 BBDF(6) 1.09196e-5 0.0089108066 ABBDF 6.29854e-6 0.0082606264 $10^{-4}$ BBDF(4) 7.07097e-12 0.0145041159 BBDF(5) 3.99341e-11 0.01477376 BBDF(6) 5.75778e-11 0.01497369 ABBDF 2.51638e-11 0.014267131 $10^{-6}$ BBDF(4) 1.19693e-12 0.331891367 BBDF(5) 1.27653e-12 0.35464837 BBDF(6) 1.54599e-12 0.366591181 ABBDF 1.32749e-12 0.3299601339 $10^{-8}$ BBDF(4) 5.70600e-11 32.31383741 BBDF(5) 6.27814e-11 33.986499 BBDF(6) 8.38467e-11 35.61432122 ABBDF 6.87739e-11 32.222682490
 $H$ Method MAXE TIME $10^{-2}$ BBDF(4) 5.34238e-6 0.008280686 BBDF(5) 8.60443e-6 0.008490485 BBDF(6) 1.09196e-5 0.0089108066 ABBDF 6.29854e-6 0.0082606264 $10^{-4}$ BBDF(4) 7.07097e-12 0.0145041159 BBDF(5) 3.99341e-11 0.01477376 BBDF(6) 5.75778e-11 0.01497369 ABBDF 2.51638e-11 0.014267131 $10^{-6}$ BBDF(4) 1.19693e-12 0.331891367 BBDF(5) 1.27653e-12 0.35464837 BBDF(6) 1.54599e-12 0.366591181 ABBDF 1.32749e-12 0.3299601339 $10^{-8}$ BBDF(4) 5.70600e-11 32.31383741 BBDF(5) 6.27814e-11 33.986499 BBDF(6) 8.38467e-11 35.61432122 ABBDF 6.87739e-11 32.222682490
Comparison between the ABBDF and BBDF for solving Problem 3.2
 $H$ Method MAXE TIME $10^{-2}$ BBDF(4) 6.13882e+1 0.008227781 BBDF(5) 2.14167e-1 0.0082600851 BBDF(6) 2.96326e+0 0.00828741 ABBDF 1.44159e-2 0.008064681 $10^{-4}$ BBDF(4) 7.76160e-8 0.01543074 BBDF(5) 9.48795e-8 0.016383168 BBDF(6) 1.11123e-7 0.0171090074 ABBDF 7.76160e-8 0.0142214673 $10^{-6}$ BBDF(4) 1.93315e-11 2.97284740 BBDF(5) 1.07853e-11 3.12728618 BBDF(6) 2.66430e-11 3.8798845185 ABBDF 1.93315e-11 2.31853289 $10^{-8}$ BBDF(4) 2.35567e-9 84.36481221 BBDF(5) 4.41047e-10 85.76521578 BBDF(6) 2.59949e-9 87.69974205 ABBDF 2.35567e-9 84.36227574
 $H$ Method MAXE TIME $10^{-2}$ BBDF(4) 6.13882e+1 0.008227781 BBDF(5) 2.14167e-1 0.0082600851 BBDF(6) 2.96326e+0 0.00828741 ABBDF 1.44159e-2 0.008064681 $10^{-4}$ BBDF(4) 7.76160e-8 0.01543074 BBDF(5) 9.48795e-8 0.016383168 BBDF(6) 1.11123e-7 0.0171090074 ABBDF 7.76160e-8 0.0142214673 $10^{-6}$ BBDF(4) 1.93315e-11 2.97284740 BBDF(5) 1.07853e-11 3.12728618 BBDF(6) 2.66430e-11 3.8798845185 ABBDF 1.93315e-11 2.31853289 $10^{-8}$ BBDF(4) 2.35567e-9 84.36481221 BBDF(5) 4.41047e-10 85.76521578 BBDF(6) 2.59949e-9 87.69974205 ABBDF 2.35567e-9 84.36227574
Comparison between the ABBDF and mathod in [13] for solving Problem 3.3
 $H$ Method MAXE $10^{-3}$ BBDF(5) 7.43088e-3 BDF 4.97642e+43 ode15s 6.30000e-3 ABBDF 2.02095e-3 $10^{-5}$ BBDF(5) 7.68218e-4 BDF 7.88641e-4 ode15s 6.40000e-3 ABBDF 6.22269e-4 $10^{-7}$ BBDF(5) 8.09607e-8 BDF 8.39232e-8 ode15s 4.30000e-5 ABBDF 6.49214e-8
 $H$ Method MAXE $10^{-3}$ BBDF(5) 7.43088e-3 BDF 4.97642e+43 ode15s 6.30000e-3 ABBDF 2.02095e-3 $10^{-5}$ BBDF(5) 7.68218e-4 BDF 7.88641e-4 ode15s 6.40000e-3 ABBDF 6.22269e-4 $10^{-7}$ BBDF(5) 8.09607e-8 BDF 8.39232e-8 ode15s 4.30000e-5 ABBDF 6.49214e-8
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