World Academy of Science, Engineering and Technology
Published February 27, 2011 | Version 15645
Journal article Open

An eighth order Backward Differentiation Formula with Continuous Coefficients for Stiff Ordinary Differential Equations

Description

A block backward differentiation formula of uniform order eight is proposed for solving first order stiff initial value problems (IVPs). The conventional 8-step Backward Differentiation Formula (BDF) and additional methods are obtained from the same continuous scheme and assembled into a block matrix equation which is applied to provide the solutions of IVPs on non-overlapping intervals. The stability analysis of the method indicates that the method is L0-stable. Numerical results obtained using the proposed new block form show that it is attractive for solutions of stiff problems and compares favourably with existing ones.

Files

15645.pdf

Files (218.6 kB)

Name Size
md5:aee780d376ecff52449d8dbf7c8a5014
218.6 kB Preview Download

Additional details

References

  • C. F. Curtiss and J. O. Hirschfelder, "Integration of stiff equations," Proceedings of the National Academy of Sciences of the United States of America, vol. 38, no. 3, p. 235, 1952.
  • E. Hairer, G. Wanner, and S. O. service), Solving ordinary differential equations II. New York: Springer Berlin, 2010.
  • P. Henrici, Discrete Variable Methods in ODEs. New York: John Wiley, 1962.
  • G. G. Dahlquist, "A special stability problem for linear multistep methods," BIT Numerical Mathematics, vol. 3, no. 1, p. 27-43, 1963.
  • J. C. Butcher and J. Wiley, Numerical methods for ordinary differential equations. New York: Wiley Online Library, 2008.
  • C. W. Gear, "Hybrid methods for initial value problems in ordinary differential equations," Journal of the Society for Industrial and Applied Mathematics: Series B, Numerical Analysis, vol. 2, no. 1, p. 69-86, 1965.
  • W. B. Gragg and H. J. Stetter, "Generalized multistep predictor-corrector methods," Journal of the ACM (JACM), vol. 11, no. 2, p. 188-209, 1964.
  • J. C. Butcher, "A modified multistep method for the numerical integration of ordinary differential equations," Journal of the ACM (JACM), vol. 12, no. 1, p. 124-135, 1965.
  • O. Akinfenwa, S. Jator, and N. Yao, "Implicit two step continuous hybrid block methods with four Off-Steps points for solving stiff ordinary differential equation," in Proceedings of the International Conference on Computational and Applied Mathematics, Bangkok ,Thailand, 2011. [10] J. B. Keiper and C. W. Gear, "The analysis of generalized backwards difference formula methods applied to hessenberg form differentialalgebraic equations," SIAM journal on numerical analysis, vol. 28, no. 3, p. 833-858, 1991. [11] W. H. Enright, "Second derivative multistep methods for stiff ordinary differential equations," SIAM Journal on Numerical Analysis, vol. 11, no. 2, p. 321-331, 1974. [12] W. H. Enright, "Continuous numerical methods for ODEs with defect control* 1," Journal of computational and applied mathematics, vol. 125, no. 1-2, p. 159-170, 2000. [13] J. R. Cash, "On the exponential fitting of composite, multiderivative linear multistep methods," SIAM Journal on Numerical Analysis, vol. 18, no. 5, p. 808-821, 1981. [14] L. Brugnano and D. Trigiante, Solving differential problems by multistep initial and boundary value methods. Amsterdam: Gordon a. Breach Science Publ., 1998. [15] M. K. Jain, Numerical solution of differential equations. Wiley Eastern, 1984. [16] J. B. Rosser, "A Runge-Kutta for all seasons," Siam Review, vol. 9, no. 3, p. 417-452, 1967. [17] D. Sarafyan, "Multistep methods for the numerical solution of ordinary differential equations made self-starting," WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER, Math. Res. Center, Madison, Tech. Rep. 495, 1965. [18] L. F. Shampine and H. A. Watts, "Block implicit one-step methods," Math. comp, vol. 23, p. 731-740, 1969. [19] S. O. Fatunla, "Block methods for second order IVPs, intern," J. Compt. Maths, vol. 41, p. 55-63, 1991. [20] P. Chartier, "L-stable parallel one-block methods for ordinary differential equations," SIAM journal on numerical analysis, vol. 31, no. 2, p. 552-571, 1994. [21] L. C. BAKER, Tools for Scientist and Engineers. New York: McGraw- Hill, 1989. [22] M. M. Stabrowski, "An efficient algorithm for solving stiff ordinary differential equations," Simulation Practice and Theory, vol. 5, no. 4, p. 333-344, 1997. [23] J. Vigo-Aguiar and H. Ramos, "A family of a-stable Runge-Kutta collocation methods of higher order for initial-value problems," IMA journal of numerical analysis, vol. 27, no. 4, p. 798, 2007. [24] G. Hall, Modern Numerical Methods for Ordinary Differential Equations: Edited by G. Hall and JM Watt. Oxford, UK: Clarendon Press, 1976. [25] P. Kaps, G. Dahlquist, and R. Jeltsch, Rosenbrock-type methods in Numerical Methods for Stiff Initial Value Problems. Germany: fur Geometrie und Praktische Mathematik der RWTH Aachen, 1981. [26] X. Y. Wu and J. L. Xia, "Two low accuracy methods for stiff systems," Applied mathematics and computation, vol. 123, no. 2, p. 141-153, 2001. [27] L. rong Chen and D. gui Liu, "Parallel rosenbrock methods for solving stiff systems in real-time simulation," Journal of Computational Mathematics, vol. 18, pp. 375-386, 2000.