Fifth Order Variable Step Block Backward Differentiation Formulae for Solving Stiff ODEs
The implicit block methods based on the backward
differentiation formulae (BDF) for the solution of stiff initial value
problems (IVPs) using variable step size is derived. We construct a
variable step size block methods which will store all the coefficients
of the method with a simplified strategy in controlling the step size
with the intention of optimizing the performance in terms of
precision and computation time. The strategy involves constant,
halving or increasing the step size by 1.9 times the previous step size.
Decision of changing the step size is determined by the local
truncation error (LTE). Numerical results are provided to support the
enhancement of method applied.
[1] L.G. Birta and O. Abou-Rabiaa, "Parallel block predictor-corrector
methods for odes,",IEEE Transactions on Computers, vol. C-36(3), pp.
299-311, 1987.
[2] K. Burrage, "Efficient block predictor-corrector methods with a small
number of corrections," J. of Comp. and App. Mat,. vol. 45, pp. 139-150,
1993.
[3] J.R. Cash, "On the integration of stiff systems of odes using extended
backward differentiation formulae," Numer. Math,.vol. 34, pp. 235-246,
1980.
[4] J.R. Cash, "The integration of stiff initial value problems in odes using
modified extended backward differentiation formulae," Comput. Math.
Appl., vol. 9, pp. 645-660, 1983.
[5] M.T. Chu, and H. Hamilton, "Parallel solution of odes by multi-block
methods," Siam J. Sci. Stat. Comput., vol. 8(1), pp. 342-353, 1987.
[6] S.O. Fatunla, "Block methods for second order odes," Intern. J.
Computer Math., vol. 40, pp. 55-63, 1990.
[7] C.W. Gear, "Numerical initial value problems in ordinary differential
equations," COMM. ACM., vol. 14, pp. 185-190, 1971.
[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, vol. 21, no.3, pp. 508-520, 2008.
[9] Z.B. Ibrahim, K.I. Othman and M.B. Suleiman, "Variable
stepsize block backward differentiation formula for solving stiff odes,"
Proceedings of World Congress on Engineering 2007, LONDON, U.K.,
vol. 2, pp. 785-789, 2007.
[10] Z.B. Ibrahim, M.B. Suleiman and K.I. Othman, "Implicit r-point block
backward differentiation formula for solving first- order stiff odes,"
Applied Mathematics and Computation, vol. 186, pp. 558-565, 2007.
[11] Z.B. Ibrahim, "Block Multistep Methods For Solving Ordinary
Differential Equations," Ph. D. Thesis, Universiti Putra Malaysia,
Selangor, 2006.
[12] P. Kaps and G. Wanner, "A study of rosenbrock-type methods of high
order," Numer. Math., vol. 38, pp. 279-298, 1981.
[13] J.D. Lambert, Numerical Methods for Ordinary Differential Equations:
The Initial Value Problems, John Wiley & Sons, New York 1991.
[1] L.G. Birta and O. Abou-Rabiaa, "Parallel block predictor-corrector
methods for odes,",IEEE Transactions on Computers, vol. C-36(3), pp.
299-311, 1987.
[2] K. Burrage, "Efficient block predictor-corrector methods with a small
number of corrections," J. of Comp. and App. Mat,. vol. 45, pp. 139-150,
1993.
[3] J.R. Cash, "On the integration of stiff systems of odes using extended
backward differentiation formulae," Numer. Math,.vol. 34, pp. 235-246,
1980.
[4] J.R. Cash, "The integration of stiff initial value problems in odes using
modified extended backward differentiation formulae," Comput. Math.
Appl., vol. 9, pp. 645-660, 1983.
[5] M.T. Chu, and H. Hamilton, "Parallel solution of odes by multi-block
methods," Siam J. Sci. Stat. Comput., vol. 8(1), pp. 342-353, 1987.
[6] S.O. Fatunla, "Block methods for second order odes," Intern. J.
Computer Math., vol. 40, pp. 55-63, 1990.
[7] C.W. Gear, "Numerical initial value problems in ordinary differential
equations," COMM. ACM., vol. 14, pp. 185-190, 1971.
[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, vol. 21, no.3, pp. 508-520, 2008.
[9] Z.B. Ibrahim, K.I. Othman and M.B. Suleiman, "Variable
stepsize block backward differentiation formula for solving stiff odes,"
Proceedings of World Congress on Engineering 2007, LONDON, U.K.,
vol. 2, pp. 785-789, 2007.
[10] Z.B. Ibrahim, M.B. Suleiman and K.I. Othman, "Implicit r-point block
backward differentiation formula for solving first- order stiff odes,"
Applied Mathematics and Computation, vol. 186, pp. 558-565, 2007.
[11] Z.B. Ibrahim, "Block Multistep Methods For Solving Ordinary
Differential Equations," Ph. D. Thesis, Universiti Putra Malaysia,
Selangor, 2006.
[12] P. Kaps and G. Wanner, "A study of rosenbrock-type methods of high
order," Numer. Math., vol. 38, pp. 279-298, 1981.
[13] J.D. Lambert, Numerical Methods for Ordinary Differential Equations:
The Initial Value Problems, John Wiley & Sons, New York 1991.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:55463", author = "S.A.M. Yatim and Z.B. Ibrahim and K.I. Othman and F. Ismail", title = "Fifth Order Variable Step Block Backward Differentiation Formulae for Solving Stiff ODEs", abstract = "The implicit block methods based on the backward
differentiation formulae (BDF) for the solution of stiff initial value
problems (IVPs) using variable step size is derived. We construct a
variable step size block methods which will store all the coefficients
of the method with a simplified strategy in controlling the step size
with the intention of optimizing the performance in terms of
precision and computation time. The strategy involves constant,
halving or increasing the step size by 1.9 times the previous step size.
Decision of changing the step size is determined by the local
truncation error (LTE). Numerical results are provided to support the
enhancement of method applied.", keywords = "Backward differentiation formulae, block backwarddifferentiation formulae, stiff ordinary differential equation, variablestep size.", volume = "4", number = "2", pages = "250-3", }