Abstract
This paper presents a family of hybrid linear multistep methods (LMMs) with second derivative term for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The methods are stiffly stable for step number k ≤ 7.
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References
Butcher, J.C., Modified Multistep Method for the Numerical Integration of Ordinary Differential Equations, J. Assoc. Comput. Mach., 1965, vol. 12, pp.124–135.
Butcher, J.C., The Numerical Analysis of Ordinary Differential Equation: Runge Kutta and General Linear Methods, Chichester: Wiley, 1987.
Butcher, J.C., Some New Hybrid Methods for IVPs, in Computational Ordinary Differential Equations, Cash, J.R. and GladWell, I., Eds., Oxford: Clarendon Press, 1992, pp.29–46.
Butcher, J.C. and O’Sullivan, A.E., Nordsieck Methods with an Off-Step Point, Num. Algor., 2002, vol. 31, pp.87–101.
Gragg, W.B. and Stetter, H.J., Generalized Multistep Predictor-Corrector Methods, J. Assoc. Comput. Mach., 1964, vol. 11, pp.188–209.
Dahlquist, G., On Stability and Error Analysis for Stiff Nonlinear Problems, part 1, Stockholm: Dept. of Information Processing, Computer Science, Royal Inst. of Technology, 1975, Rep. TRITA-NA-7508.
Enright, W.H., Second Derivative Multistep Methods for Stiff ODEs, SIAM. J. Num. Anal., 1974, vol. 11,iss. 2, pp.321–331.
Enright, W.H., Hull, T.E, and Lindberg, B., Comparing Numerical Methods for Stiff Systems of ODEs, BIT Num. Math., 1975, vol. 15, no. 1, pp.10–48.
Fatunla, S.O., Numerical Methods for Initial Value Problems in ODEs, New York: Academic Press, 1978.
Gear, C.W., The Automatic Integration of Stiff ODEs, Information Processing 68: Proc. IFIP Congress, Edinburgh, 1968, Morrell, A.J.H., Ed., Amsterdam: North-Holland, 1969, pp.187–193.
Gear, C.W., Algorithm 407: DIFSUB for Solution of ODEs, Comm. ACM, 1971, vol. 14,iss. 3, pp.185–190.
Gear, C.W., Numerical Initial Value Problems in ODEs, Englewood Cliffs, N.J., USA: Prentice-Hall, 1971.
Higham, D.J. and Higham, N.J., Matlab Guide, Philadelphia: SIAM, 2000.
Ikhile, M.N.O. and Okuonghae, R.I., Stiffly Stable Continuous Extension of Second Derivative LMM with an Off-Step Point for IVPs in ODEs, J. Nig. Assoc. Math. Phys., 2007, vol. 11, pp.175–190.
Kohfeld, J.J. and Thompson, G.T., Multistep Methods with Modified Predictors and Correctors, J. Assoc. Comput. Mach., 1967, vol. 14, pp.155–166.
Lambert, J.D., Numerical Methods for Ordinary Differential Systems. The Initial Value Problems, Chichester: Wiley, 1991.
Lambert, J.D., Computational Methods for Ordinary Differential Systems. The Initial Value Problems, Chichester: Wiley, 1973.
Nevanlinna, O., On the Numerical Integration of Nonlinear IVPs by Linear Multistep Methods, BIT Num. Math., 1977, vol. 17, pp.58–71.
Owren, B. and Zennaro, M., Order Barriers for Continuous Explicit Runge-Kutta Methods, Math. Comput., 1991, vol. 56, pp.645–661.
Okuonghae, R.I., Stiffly Stable Second Derivative Continuous LMM for IVPs in ODEs, PhD Thesis, Nigeria, Benin City: Dept. of Math. University of Benin, 2008.
Okuonghae, R.I., Ogunleye, S.O., and Ikhile, M.N.O., Some Explicit General Linear Methods for IVPs in ODEs, J. Algor. Comp. Technol., 2013, vol. 7, no. 1, pp.41–63.
Okuonghae, R.I., Variable Order Explicit Second Derivative General Linear Methods, Comp. Appl. Math., 2014, vol. 33, pp.243–255.
Okuonghae, R.I. and Ikhile, M.N.O., On the Construction of High Order A(α)-Stable Hybrid Linear Multistep Methods for Stiff IVPs and ODEs, Num. An. Appl., 2012, vol. 5, no. 3, pp.231–241.
Okuonghae, R.I. and Ikhile, M.N.O., Second Derivative General Linear Methods, Num. Algor., 2014, vol. 67,iss. 3, pp.637–654.
Okuonghae, R.I. and Ikhile, M.N.O., A Class of Hybrid Linear Multistep Methods with A(α)-Stability Properties for Stiff IVPs in ODEs, J. Num. Math., 2013, vol. 21, no. 2, pp.157–172.
Okuonghae, R.I. and Ikhile, M.N.O., A-Stable High Order Hybrid Linear Multistep Methods for Stiff Problems, J. Algor. Comp. Technol., 2014, vol. 8, no. 4, pp.441–469.
Widlund, O., A Note on Unconditionally Stable Linear Multistep Methods, BIT Num. Math., 1967, vol. 7, pp.65–70.
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Original Russian Text © R.I. Okuonghae, M.N.O. Ikhile, 2015, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2015, Vol. 18, No. 3, pp. 299–311.
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Okuonghae, R.I., Ikhile, M.N.O. Stiffly stable second derivative linear multistep methods with two hybrid points. Numer. Analys. Appl. 8, 248–259 (2015). https://doi.org/10.1134/S1995423915030052
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DOI: https://doi.org/10.1134/S1995423915030052