| Title:
|
The Numerical Solution of Stiff IVPs in ODEs Using Modified Second Derivative BDF (English) |
| Author:
|
Okuonghae, R. I. |
| Author:
|
Ikhile, M. N. O. |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
51 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
51-77 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper considers modified second derivative BDF (MSD-BDF) for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The methods are A$(\alpha )$-stable for step length $k\le 7$. (English) |
| Keyword:
|
second derivative BDF |
| Keyword:
|
collocation and interpolation |
| Keyword:
|
initial value problem |
| Keyword:
|
stiff stability |
| Keyword:
|
boundary locus |
| MSC:
|
34A34 65L06 65L20 |
| MSC:
|
65L04 |
| MSC:
|
65L05 |
| MSC:
|
65L06 |
| idZBL:
|
Zbl 06204921 |
| idMR:
|
MR3060009 |
| . |
| Date available:
|
2012-06-25T08:23:41Z |
| Last updated:
|
2014-03-12 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142874 |
| . |
| Reference:
|
[1] Butcher, J. C.: A modified multistep method for the numerical integration of ordinary differential equations. J. Assoc. Comput. Mach. 12 (1965), 124–135. Zbl 0125.07102, MR 0178573, 10.1145/321250.321261 |
| Reference:
|
[2] Butcher, J. C.: The Numerical Analysis of Ordinary Differential Equation: Runge Kutta and General Linear Methods. Wiley, Chichester, 1987. MR 0878564 |
| Reference:
|
[3] Butcher, J. C.: Some new hybrid methods for IVPs. In: Cash, J.R., Gladwell, I. (eds) Computational Ordinary Differential Equations Clarendon Press, Oxford, 1992, 29–46. MR 1387122 |
| Reference:
|
[4] Butcher, J. C.: High Order A-stable Numerical Methods for Stiff Problems. Journal of Scientific Computing 25 (2005), 51–66. Zbl 1203.65106, MR 2231942, 10.1007/s10915-004-4632-8 |
| Reference:
|
[5] 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 |
| Reference:
|
[6] Butcher, J. C.: Numerical Methods for Ordinary Differential Equations. sec. edi., Wiley, Chichester, 2008. Zbl 1167.65041, MR 2401398 |
| Reference:
|
[7] Butcher, J. C.: General linear methods for ordinary differential equations. Mathematics and Computers in Simulation 79 (2009), 1834–1845. Zbl 1159.65333, MR 2494513, 10.1016/j.matcom.2007.02.006 |
| Reference:
|
[8] Butcher, J. C.: Trees and numerical methods for ordinary differential equations. Numerical Algorithms 53 (2010), 153–170. Zbl 1184.65072, MR 2600925, 10.1007/s11075-009-9285-0 |
| Reference:
|
[9] Butcher, J. C., Hojjati, G.: Second derivative methods with RK stability. Numer. Algorithms 40 (2005), 415–429. Zbl 1084.65069, MR 2191975, 10.1007/s11075-005-0413-1 |
| Reference:
|
[10] Butcher, J. C., Rattenbury, N.: ARK Methods for Stiff Problems. Appl. Numer. Math. 53 (2005), 165–181. Zbl 1070.65059, MR 2128520, 10.1016/j.apnum.2004.09.033 |
| Reference:
|
[11] Coleman, J. P., Duxbury, S. C.: Mixed collocation methods for $y^{\prime \prime }= f(x, y)$. Research Report NA-99/01, 1999 Dept. Math. Sci., University of Durham, J. Comput. Appl. (2000), 47–75. MR 1806107 |
| Reference:
|
[12] Dahlquist, G.: On stability and error analysis for stiff nonlinear problems. Part 1. Report No TRITA-NA-7508, Dept. of Information processing, Computer Science, Royal Inst. of Technology, Stockholm, 1975. |
| Reference:
|
[13] Enright, W. H.: Second derivative multistep methods for stiff ODEs. SIAM J. Num. Anal. 11 (1974), 321–331. MR 0351083, 10.1137/0711029 |
| Reference:
|
[14] Enright, W. H.: Continuous numerical methods for ODEs with defect control. J. Comput. Appl. Math. 125 (2000), 159–170. Zbl 0982.65078, MR 1803189, 10.1016/S0377-0427(00)00466-0 |
| Reference:
|
[15] Enright, W. H., Hull, T. E., Linberg, B.: Comparing numerical Methods for Stiff of ODEs systems. BIT 15 (1975), 1–48. 10.1007/BF01932994 |
| Reference:
|
[16] Fatunla, S. O.: Numerical Methods for Initial Value Problems in ODEs. Academic Press, New York, 1978. |
| Reference:
|
[17] Forrington, C. V. D.: Extensions of the predictor-corrector method for the solution of systems of ODEs. Comput. J. 4 (1961), 80–84. 10.1093/comjnl/4.1.80 |
| Reference:
|
[18] 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 |
| Reference:
|
[19] Gear, C. W.: The automatic integration of ODEs. Comm. ACM 14 (1971), 176–179. MR 0388778, 10.1145/362566.362571 |
| Reference:
|
[20] Gragg, W. B., Stetter, H. J.: Generalized multistep predictor corrector methods. J. Assoc. Comput. Mach. 11 (1964), 188–209. Zbl 0168.13803, MR 0161476, 10.1145/321217.321223 |
| Reference:
|
[21] Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, 1996. Zbl 0859.65067, MR 1439506 |
| Reference:
|
[22] Higham, J. D., Higham, J. N.: Matlab Guide. SIAM, Philadelphia, 2000. Zbl 0953.68642, MR 1787308 |
| Reference:
|
[23] 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. |
| Reference:
|
[24] Kohfeld, J. J., Thompson, G. T.: Multistep methods with modified predictors and correctors. J. Assoc. Comput. Mach. 14 (1967), 155–166. Zbl 0173.17906, MR 0242375, 10.1145/321371.321383 |
| Reference:
|
[25] Lambert, J. D.: Numerical Methods for Ordinary Differential Systems. The Initial Value Problems. Wiley, Chichester, 1991. MR 1127425 |
| Reference:
|
[26] Lambert, J. D.: Computational Methods for Ordinary Differential Systems. The Initial Value Problems. Wiley, Chichester, 1973. |
| Reference:
|
[27] 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. |
| Reference:
|
[28] Okuonghae, R. I.: A class of Continuous hybrid LMM for stiff IVPs in ODEs. Scientific Annals of AI. I. Cuza University of Iasi, (2010), Accepted for publication. |
| Reference:
|
[29] Okuonghae, R. I., Ikhile, M. N. O.: A continuous formulation of $A(\alpha )$-stable second derivative linear multistep methods for stiff IVPs and ODEs. J. of Algorithms and Comp. Technology, (2011), Accepted for publication. MR 2964215 |
| Reference:
|
[30] Okuonghae, R. I., Ikhile, M. N. O.: $A(\alpha )$-stable linear multistep methods for stiff IVPs and ODEs. Acta. Univ. Palacki. Olomuc., Fac. rer. nat., Math. 50 (2011), 73–90. MR 2920700 |
| Reference:
|
[31] Selva, M., Arevalo, C., Fuherer, C.: A Collocation formulation of multistep methods for variable step-size extensions. Appl. Numer. Math. 42 (2002), 5–16. MR 1921325, 10.1016/S0168-9274(01)00138-6 |
| Reference:
|
[32] Widlund, O.: A note on unconditionally stable linear multistep methods. BIT 7 (1967), 65–70. Zbl 0178.18502, MR 0215533, 10.1007/BF01934126 |
| . |
