Geodesic
On the construction of high order $A(\alpha)$-stable hybrid linear multistep methods for stiff IVPs in ODEs
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 15 (2012) no. 3, pp. 281-292.

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In this paper, we present a class of $A(\alpha)$-stable hybrid linear multistep methods for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The method considered uses a second derivative like the Enright's second derivative linear multistep methods for stiff IVPs in ODEs. 
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R. I. Okuonghae; M. N. O. Ikhile. On the construction of high order $A(\alpha)$-stable hybrid linear multistep methods for stiff IVPs in ODEs. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 15 (2012) no. 3, pp. 281-292. http://geodesic.mathdoc.fr/item/SJVM_2012_15_3_a4/

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

[2] Butcher J. C., “High order $A$-stable numerical methods for stiff problems”, J. of Scientific Computing, 25:1 (2005), 51–66 | DOI | MR | Zbl

[3] Butcher J. C., Hojjati G., “Second derivative methods with RK stability”, Numer. Algorithms, 40:4 (2005), 415–429 | DOI | MR | Zbl

[4] Butcher J. C., “Forty-five years of $A$-stability”, Proc. Inter. Conf. on Numer. Anal. and Appl. Math., Ser. AIP, 1048, 2008 | MR

[5] Butcher J. C., Numerical Methods for Ordinary Differential Equations, Second Edition, Wiley, Chichester, 2008 | MR | Zbl

[6] Butcher J. C., “General linear methods for ordinary differential equations”, Math. and Comput. in Simulation, 79 (2009), 1834–1845 | DOI | MR | Zbl

[7] Butcher J. C., “Trees and numerical methods for ordinary differential equations”, Numer. Algorithms, 53 (2010), 153–170 | DOI | MR | Zbl

[8] Burrage K., Tian T., “Stiffly accurate Runge–Kutta methods for stiff stochastic differential equations”, Comput. Phys. Commun., 142 (2001), 186–190 | DOI | MR | Zbl

[9] Butcher J. C., Chan D. J. L., “On the implementation of ESIRK methods for stiff IVPs”, Numer. Algorithms, 26 (2001), 210–218 | MR

[10] Dahlquist G., “A special stability problem for linear multistep methods”, BIT, 3 (1963), 27–43 | DOI | MR | Zbl

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

[12] Enright W. H., “Continuous numerical methods for ODEs with defect control”, J. Comput. Appl., 125 (2000), 159–170 | DOI | MR | Zbl

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

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

[15] Filippov S. S., Tygliyan A. V., “A class of linearly implicit numerical methods for solving stiff ordinary differential equations”, The Open Numer. Methods J., 2 (2010), 1–5 | DOI | MR

[16] Gear C. W., “The automatic integration of stiff ODEs”, Information processing 68, Proc. IFIP Congress (Edinurgh, 1968), eds. A. J. H. Morrell, Nort-Holland, Amsterdam, 1968, 187–193 | MR

[17] Gear C. W., “The automatic integration of ODEs”, Comm. of the ACM, 14 (1871), 176–179 | DOI | MR

[18] Hairer E., Wanner G., Solving Ordinary Differential Equations, v. II, Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, 1996 | MR | Zbl

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

[20] 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. Phys., 11 (2007), 175–190

[21] Ikhile M. N. O., “Coefficients for studing one-step rational schemes for IVPs in ODEs. III. Extrapolation methods”, Comp. and Math. with Appl., 47 (2004), 1463–1475 | DOI | MR | Zbl

[22] Kaps P., “Rosenbrock-type methods”, Numerical Methods for Solving Stiff Initial Value Problems, eds. Dahlquist G., Jeltsch R., Inst. für Geometrie und praktische Math. (IGPM) der RWTH Aachen, Aachen, Germany, 1981, Bericht No 9

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

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

[25] Wolfram Mathematica, vs 6.0, Wolfram Research Inc.

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

[27] Robertson H. H., “The solution of a set of reaction rate equations”, Numer. Anal.: an Introduction, ed. J. Walsh, Academ. Press, London, 1966, 178–182

[28] Selva M., Arevalo C., Fuherer C., “A collocation formulation of multistep methods for variable step-size extensions”, Appl. Numer. Math., 42 (2002), 5–16 | DOI | MR | Zbl

[29] Shampine L. F., Watts H. A., “Block implicit one-step methods”, Math. of Comp., 23 (1969), 731–740 | DOI | MR | Zbl

[30] Sirisena U., Onumanyi P., Chollon J. P., “Continuous hybrid methods through multistep collocation”, ABACUS, 28 (2002), 58–66

[31] Tischer P. E., Gupta G. K., Some new Cyclic Linear Multistep Formulas for Stiff System, Tech. Report No 40, Nov., Dept. of Comp. Science, Monash University, Clayton Victoria, Australia, 1983

[32] Voss D. A., “Fourth-order parallel Rosebrock methods for stiff systems”, Math. and Comp. Modelling J., 40 (2004), 1193–1198 | DOI | MR | Zbl

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

[34] Zarina B. I., Khairil I. O., Mohammed S., “Variable step block backward differentiation formula for solving first order stiff ODEs”, Proc. WCE, 2166, WCE, London, 2007, 785–789