Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 32131
Implicit Two Step Continuous Hybrid Block Methods with Four Off-Steps Points for Solving Stiff Ordinary Differential Equation

Authors: O. A. Akinfenwa, N.M. Yao, S. N. Jator


In this paper, a self starting two step continuous block hybrid formulae (CBHF) with four Off-step points is developed using collocation and interpolation procedures. The CBHF is then used to produce multiple numerical integrators which are of uniform order and are assembled into a single block matrix equation. These equations are simultaneously applied to provide the approximate solution for the stiff ordinary differential equations. The order of accuracy and stability of the block method is discussed and its accuracy is established numerically.

Keywords: Collocation and Interpolation, Continuous HybridBlock Formulae, Off-Step Points, Stability, Stiff ODEs.

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1948


[1] K. E. Atkinson, An introduction to numerical analysis, 2nd edition John Wiley and Sons, New York, 1989.
[2] L. Brugnano and D. Trigiante, Solving Differential Problems by Multitep Initial and Boundary Value Methods, Gordon and Breach Science Publishers, Amsterdam, 1998.
[3] J. C. Butcher, A modified multistep method for the numerical integration of ordinary differential equations, J. Assoc. Comput. Mach. 12 (1965),pp. 124-135.
[4] M.T. Chu & H. Hamilton, Parallel solution of ODE-s by multi-block methods, SIAM J. Sci. Stat. Comput. 8, 342-353, (1987).
[5] Dahlquist, G. A Special Stability Problem for Linear Multistep Methods, BIT, 3,1963, pp.27-43.
[6] S. O. Fatunla, Block methods for second order IVPs, Intern. J. Comput. Math. 41 (1991), pp. 55 - 63.
[7] C. W. Gear, Hybrid methods for initial value problems in ordinary differential equations, SIAM J. Numer. Anal. 2 (1965), pp. 69-86.
[8] I. Gladwell and D. K. Sayers, Eds. Computational techniques for ordinary differential equations, Academic Press, New York, 1976.
[9] W. Gragg and H. J. Stetter, Generalized multistep predictor-corrector methods, J.Assoc. Comput. Mach., 11 (1964), pp. 188-209.
[10] G. K. Gupta, Implementing second-derivative multistep methods using Nordsieck polynomial representation, Math. Comp. 32 (1978), pp.13-18.
[11] P. Henrici, Discrete Variable Methods in ODEs, John Wiley, New York, 1962.
[12] S.N Jator, On The Hybrid Method With Three-off Step Points For Initial Value Problems, International Journal Of Mathematics Education in Science and Technology ,1464- 5211,Volume 41 Issue 1(2010) , pp.110-118.
[13] J. J. Kohfeld and G. T. Thompson, Multistep methods with modified predictors and correctors, J. Assoc. Comput. Mach., 14 (1967), pp. 155- 166.
[14] J. D. Lambert Computational methods in ordinary differential equations, John Wiley, New York, 1973.
[15] I. Lie and S. P. Norsett,1989, Superconvergence for Multistep Collocation, Math Comp. 52 (1989) pp. 65 79.
[16] W. E. Milne, Numerical solution of differential equations, John Wiley and Sons, 1953.
[17] P. Onumanyi, D. O. Awoyemi, S. N. Jator, and U. W. Sirisena, 1994,New linear mutlistep methods with continuous coefficients for firstorder initial value problems, J. Nig. Math. Soc. 13 (1994), pp. 37- 51.
[18] J. D. Rosser, A Runge-kutta for all seasons, SIAM, Rev., 9 (1967), pp.417-452.
[19] D. Sarafyan Multistep methods for the numerical solution of ordinary differential equations made self-starting, Tech. Report 495, Math. Res. Center, Madison (1965).
[20] L. F. Shampine and Watts, H. A., "Block Implicit One-Step Methods", Math. Comp. 23, 1969, pp. 731-740.