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Keywords:
second derivative method; collocation and interpolation; initial value problem; stiff stability; boundary locus
second derivative method; collocation and interpolation; initial value problem; stiff stability; boundary locus
Summary:
In this paper, a class of A($\alpha $)-stable linear multistep formulas for stiff initial value problems (IVPs) in ordinary differential equations (ODEs) is developed. The boundary locus of the methods shows that the schemes are A-stable for step number $k\le 3$ and stiffly stable for $k=4, 5$ and $6$. Some numerical results are reported to illustrate the method.
References:
[1] Butcher, J. C.: The Numerical Analysis of Ordinary Differential Equation: Runge Kutta and General Linear Methods. Wiley, Chichester, 1987. MR 0878564
[2] Butcher, J. C.: High Order A-stable Numerical Methods for Stiff Problems. Journal of Scientific Computing 25 (2005), 51–66. MR 2231942 | Zbl 1203.65106
[3] Butcher, J. C., Hojjati, G.: Second derivative methods with RK stability. Numer. Algorithms 40 (2005), 415–429. DOI 10.1007/s11075-005-0413-1 | MR 2191975 | Zbl 1084.65069
[4] Butcher, J. C.: Forty-five years of A-stability. In: Numerical Analysis and Applied Mathematics: International Conference on Numerical Analysis and Applied Mathematics 2008. AIP Conference Proceedings 1048 (2008). MR 2598780
[5] Butcher, J. C.: Numerical Methods for Ordinary Differential Equations. sec. edi., Wiley, Chichester, 2008. MR 2401398 | Zbl 1167.65041
[6] Butcher, J. C.: General linear methods for ordinary differential equations. Mathematics and Computers in Simulation 79 (2009), 1834–1845. DOI 10.1016/j.matcom.2007.02.006 | MR 2494513 | Zbl 1159.65333
[7] Butcher, J. C.: Trees and numerical methods for ordinary differential equations. Numerical Algorithms 53 (2010), 153–170. DOI 10.1007/s11075-009-9285-0 | MR 2600925 | Zbl 1184.65072
[8] Dahlquist, G.: A special stability problem for linear multistep methods. BIT 3 (1963), 27–43. DOI 10.1007/BF01963532 | MR 0170477 | Zbl 0123.11703
[9] Enright, W. H.: Second derivative multistep methods for stiff ODEs. SIAM J. Num. Anal. 11 (1974), 321–331. DOI 10.1137/0711029 | MR 0351083
[10] Enright, W. H.: Continuous numerical methods for ODEs with defect control. J. Comput. Appl. Math. 125 (2000), 159–170. DOI 10.1016/S0377-0427(00)00466-0 | MR 1803189 | Zbl 0982.65078
[11] Enright, W. H., Hull, T. E., Linberg, B.: Comparing numerical Methods for Stiff of ODEs systems. BIT 15 (1975), 1–48. DOI 10.1007/BF01932994
[12] Fatunla, S. O.: Numerical Methods for Initial Value Problems in ODEs. Academic Press, New York, 1978.
[13] Gear, C. W.: The automatic integration of stiff ODEs. In: Morrell, A.J.H. (ed.) Information processing 68: Proc. IFIP Congress, Edinurgh, 1968 Nort-Holland, Amsterdam, 1968, 187–193. MR 0260180
[14] Gear, C. W.: The automatic integration of ODEs. Comm. ACM 14 (1971), 176–179. DOI 10.1145/362566.362571 | MR 0388778
[15] Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, 1996. MR 1439506 | Zbl 0859.65067
[16] Higham, J. D., Higham, J. N.: Matlab Guide. SIAM, Philadelphia, 2000. MR 1787308 | Zbl 0953.68642
[17] 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.
[18] Lambert, J. D.: Numerical Methods for Ordinary Differential Systems. The Initial Value Problems. Wiley, Chichester, 1991. MR 1127425
[19] Lambert, J. D.: Computational Methods for Ordinary Differential Systems. The Initial Value Problems. Wiley, Chichester, 1973. MR 0423815
[20] Okuonghae, R. I.: Stiffly Stable Second Derivative Continuous LMM for IVPs in ODEs. Ph.D. Thesis, Dept. of Maths. University of Benin, Benin City. Nigeria, 2008.
[21] 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 10.1016/S0168-9274(01)00138-6 | MR 1921325
[22] Widlund, O.: A note on unconditionally stable linear multistep methods. BIT 7 (1967), 65–70. DOI 10.1007/BF01934126 | MR 0215533 | Zbl 0178.18502
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