This paper considers a new class of high order hybrid linear multistep methods for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The numerical experiments shows the application of the methods on stiff problems.
ButcherJ. C., A modified multistep methods for the numerical integration of ordinary differential equations. J. ACM12, No. 1 (1965), 125–135.
2.
ButcherJ. C., High order A-stable numerical methods for stiff problems. Journal of Scientific Computing25 (2005), 51–66.
3.
ButcherJ. C., Numerical Methods for Ordinary Differential Equations.Second Edition., J. Wiley, Chichester, (2008).
4.
DahlquistG., A special stability problem for linear multistep methods, BIT, 3, (1963), pp. 27–43.
5.
EnrightW. H., Continuous numerical methods for ODEs with defect control. J. Comput. Appl. Math., 125 (2000), pp. 159–170.
6.
EnrightW. H., Second derivative multistep methods for stiff ODEs. SIAM. J. Numer. Anal.(1974), vol. 11 pp. 321–331.
7.
FatunlaS. O., Numerical Methods for Initial Value Problems in ODEs.Academic Press, New York, (1978)
8.
GraggW. B. and StetterH. J., Generalized multistep predictor-corrector methods. J. Assoc. Mach., 11, (1964), 188–209.
9.
GearC. W., Hybrid methods for initial value problems in ordinary differential equations. SIAM. J. Numer. Anal., 2, (1965), 69–86.
10.
HairerE., and WannerG., Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, (1996).
11.
IkhileM. N. O., and OkuonghaeR. I., Stiffly stable continuous extension of second derivative LMM with an off-step point for IVPs in ODEs. J. Nig. Assoc. Math. Physics.Vol.11 (2007), pp.175–190.
12.
Vigo-AguiarJ. and RamosH., A new eighth-order A-stable method for solving differential systems arising in chemical reactions. Journal of Mathematical Chemistry, Vol. 40, No. 1, July 2006.
13.
Vigo-AguiarJ. and RamosH., A family of A-stable Runge Kutta collocation methods of higher order for initial-value problems. IMA Journal of Numerical Analysis (2007) 27, 798–817
14.
KapsP., Rosenbrock-type methods. In: DahlquistGJeltschR. Eds., Numerical methods for solving stiff initial value problems. Inst. fur Geometric und praktische Math. der RWTH Aachen1981; Bericht No. 9.
15.
LieI., and NorsetS. P., Superconvergence for multistep collocation Maths. of Comp.52, No. 185, (1989), 65–79.
16.
LambertJ. D., Numerical Methods for Ordinary Differential Systems. The Initial Value Problems.Wiley, Chichester, (1991).
17.
LambertJ. D., Computational Methods for Ordinary Differential Systems. The Initial Value Problems.Wiley, Chichester, (1973).
18.
IkhileM. N. O., and OkuonghaeR. I., Stiffly stable continuous extension of second derivative LMM with an off-step point for IVPs in ODEs. J. Nig. Assoc. Math. Physics.Vol.11 (2007), pp. 175–190.
19.
OkuonghaeR. I., Stiffly Stable Second Derivative Continuous LMM for IVPs in ODEs. Ph.D Thesis. Dept. of Mathematics Benin: University of Benin, Benin city, (2008).
20.
OkuonghaeR. I., A class of continuous hybrid LMM for stiff IVPs in ODEs. Analele Stiintiifice ale Universitat II AL.I. Cuza Din Iasi (S. N.) MATEMATICA, Tomul LVIII, (2012), f.2, pp. 239–258.
21.
OkuonghaeR. I. and IkhileM. N. O., A continuous formulation of A(α)-stable second derivative linear multistep methods for stiff IVPs and ODEs. J. of Algorithms and Comp. Technology, Vol. 6, No. 1 (2011), 79–101.
22.
OkuonghaeR. I.OgunleyeS. O. and IkhileM. N. O., Some explicit general linear methods for IVPs in ODEs. J. of Algorithms and Comp. Technology, Vol. 7, No. 1 (2013), 41–63.
23.
OkuonghaeR. I. and IkhileM. N. O., A(α)-stable linear multistep methods for stiff IVPs in ODEs. Acta Univ. Placki. Olomuc., Fac. rer nat., Mathematica50, 1 (2011), 75–92.
24.
OkuonghaeR. I. and IkhileM. N. O., The numerical solution of stiff IVPs in ODEs using modified second derivative BDF. Acta Univ. Placki. Olomuc., Fac. rer. nat., Mathematica51, 1 (2012). 51–77.
25.
OkuonghaeR. I. and IkhileM. N. O., On the construction of high order A(α)-stable hybrid linear multistep methods for stiff IVPs and ODEs. Journal of Numerical Analysis and ApplicationNo. 3, Vol. 15, (2012), pp. 231–241.
26.
OkuonghaeR. I. and IkhileM.N.O., A class of hybrid linear multistep methods with A(α)-stability properties for stiff IVPs in ODEs. J. Numer. Math., Vol. 21, No.2, (2013), pp. 157–172.
27.
OkuonghaeR. I., A class of A(α)-stable numerical methods for stiff IVPs in ODEs. Journal of Numerical Analysis and Application. To appeared in Vol. 6, No 4, (2013). Springer.com.
28.
SelvaM.ArevaloC., and FuhererC., A Collocation Formulation of Multistep Methods for Variable Step-size Extensions. Appl. Numer. Math, Vol. 42 (2002). pp. 5–16.
29.
WidlundO., A Note on Unconditionally Stable Linear Multistep Methods. BIT, (1967), 7. pp. 65–70.
