4 | 0 | 0 | | 1 | | | 0 | ||
0 | 0 | | | 1 | 0 | | |||
5 | 0 | | 1 | | | 0 | |||
0 | | | 1 | 0 | | ||||
6 | - | | 1 | | | 0 | |||
- | | | 1 | 0 | |
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: |
Table 1. Coefficients for ABBDF
4 | 0 | 0 | | 1 | | | 0 | ||
0 | 0 | | | 1 | 0 | | |||
5 | 0 | | 1 | | | 0 | |||
0 | | | 1 | 0 | | ||||
6 | - | | 1 | | | 0 | |||
- | | | 1 | 0 | |
Table 2. Comparison between the ABBDF and BBDF for solving Problem 3.1
Method | MAXE | TIME | |
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 | |
| 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 | |
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 | |
| 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 |
Table 3. Comparison between the ABBDF and BBDF for solving Problem 3.2
Method | MAXE | TIME | |
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 | |
| 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 | |
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 | |
| 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 |
Table 4. Comparison between the ABBDF and mathod in [13] for solving Problem 3.3
Method | MAXE | |
BBDF(5) | 7.43088e-3 | |
BDF | 4.97642e+43 | |
ode15s | 6.30000e-3 | |
ABBDF | 2.02095e-3 | |
| BBDF(5) | 7.68218e-4 |
BDF | 7.88641e-4 | |
ode15s | 6.40000e-3 | |
ABBDF | 6.22269e-4 | |
BBDF(5) | 8.09607e-8 | |
BDF | 8.39232e-8 | |
ode15s | 4.30000e-5 | |
ABBDF | 6.49214e-8 |
[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.
![]() ![]() |