Geodesic
Stiffly stable second derivative linear multistep methods with two hybrid points
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 18 (2015) no. 3, pp. 305-317.

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This paper presents a family of hybrid linear multistep methods (LMM) with a second derivative term for the numerical solution of stiff initial value problems (IVPs) for ordinary differential equations (ODEs). The methods are stiffly stable for the step number $k\le7$. 
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R. I. Okuonghae; M. N. O. Ikhile. Stiffly stable second derivative linear multistep methods with two hybrid points. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 18 (2015) no. 3, pp. 305-317. http://geodesic.mathdoc.fr/item/SJVM_2015_18_3_a4/

[1] Butcher J. C., “A modified multistep method for the numerical integration of ODEs”, J. Assoc. Comput. Mach., 12 (1965), 124–135 | DOI | MR | Zbl

[2] Butcher J. C., The Numerical Analysis of Ordinary Differential Equation: Runge Kutta and General Linear Methods, Wiley, Chichester, 1987 | MR

[3] Butcher J. C., “Some new hybrid methods for IVPs”, Computational Ordinary Differential Equations, eds. J. R. Cash, I. GladWell, Clarendon Press, Oxford, 1992, 29–46 | MR

[4] Butcher J. C., O'Sullivan A. E., “Nordsieck methods with an off-step point”, Numerical Algorithms, 31 (2002), 87–101 | DOI | MR | Zbl

[5] Gragg W. B., Stetter H. J., “Generalized multistep predictor-corrector methods”, J. Assoc. Comput. Mach., 11 (1964), 188–209 | DOI | MR | Zbl

[6] Dahlquist G., On Stability and Error Analysis for Stiff Nonlinear Problems. Part 1, Report TRITA-NA-7508, Dept. of Information Processing, Computer Science, Royal Inst. Of Technology, Stockholm, 1975

[7] Enright W. H., “Second derivative multistep methods for stiff ODEs”, SIAM. J. Numer. Anal., 11:2 (1974), 321–331 | DOI | MR | Zbl

[8] Enright W. H., Hull T. E., Lindberg B., “Comparing numerical methods for stiff systems of ODEs”, BIT Numerical Mathematics, 15:1 (1975), 10–48 | DOI | MR | Zbl

[9] Fatunla S. O., Numerical Methods for Initial Value Problems in ODEs, Academic Press, New York, 1978

[10] Gear C. W., “The automatic integration of stiff ODEs”, Information Processing 68, Proc. IFIP Congress (Edinburgh, 1968), ed. A. J. H. Morrell, North-Holland, Amsterdam, 1969, 187–193 | MR

[11] Gear C. W., “Algorithm 407: DIFSUB for solution of ODEs”, Comm. ACM, 14:3 (1971), 185–190 | DOI | MR

[12] Gear C. W., Numerical Initial Value Problems in ODEs, Prentice-Hall, Englewood Cliffs, N.J., USA, 1971 | MR

[13] Higham D. J., Higham N. J., Matlab Guide, SIAM, Philadelphia, 2000 | MR | Zbl

[14] Ikhile M. N. O., Okuonghae R. I., “Stiffly stable continuous extension of second derivative LMM with an off-step point for IVPs in ODEs”, J. Nig. Assoc. Math. Physics, 11 (2007), 175–190

[15] Kohfeld J. J., Thompson G. T., “Multistep methods with modified predictors and correctors”, J. Assoc. Comput. Mach., 14 (1967), 155–166 | DOI | MR | Zbl

[16] Lambert J. D., Numerical Methods for Ordinary Differential Systems. The Initial Value Problems, Wiley, Chichester, 1991 | MR

[17] Lambert J. D., Computational Methods for Ordinary Differential Systems. The Initial Value Problems, Wiley, Chichester, 1973

[18] Nevanlinna O., “On the numerical integration of nonlinear IVPs by linear multistep methods”, BIT Numerical Mathematics, 17 (1977), 58–71 | DOI | MR | Zbl

[19] Owren B., Zennaro M., “Order barriers for continuous explicit Runge–Kutta methods”, Mathematics and Computation, 56 (1991), 645–661 | DOI | MR | Zbl

[20] Okuonghae R. I., Stiffly Stable Second Derivative Continuous LMM for IVPs in ODEs, PhD Thesis, Dept. of Math. University of Benin, Nigeria, Benin City, 2008

[21] Okuonghae R. I., Ogunleye S. O., Ikhile M. N. O., “Some explicit general linear methods for IVPs in ODEs”, J. of Algorithms and Comp. Technology, 7:1 (2013), 41–63 | DOI | MR | Zbl

[22] Okuonghae R. I., “Variable order explicit second derivative general linear methods”, Comp. Applied Math., 33 (2014), 243–255 | DOI | MR | Zbl

[23] Okuonghae R. I., Ikhile M. N. O., “On the construction of high order $A(\alpha)$-stable hybrid linear multistep methods for stiff IVPs and ODEs”, Numerical Analysis and Appl., 5:3 (2012), 231–241 | DOI | MR | Zbl

[24] Okuonghae R. I., Ikhile M. N. O., “Second derivative general linear methods”, Numerical Algorithms, 67:3 (2014), 637–654 | DOI | MR | Zbl

[25] Okuonghae R. I., Ikhile M. N. O., “A class of hybrid linear multistep methods with $A(\alpha)$-stability properties for stiff IVPs in ODEs”, J. Numer. Math., 21:2 (2013), 157–172 | DOI | MR | Zbl

[26] Okuonghae R. I., Ikhile M. N. O., “$A$-stable high order hybrid linear multistep methods for stiff problems”, J. of Algorithms Comp. Technology, 8:4 (2014), 441–469 | DOI | MR

[27] Widlund O., “A note on unconditionally stable linear multistep methods”, BIT Numerical Mathematics, 7 (1967), 65–70 | DOI | MR | Zbl