## The numerical solution of stiff IVPs in ODEs using modified second derivative BDF.(English)Zbl 1273.65099

Summary: This paper considers modified second derivative backward differentiation formulas (BDF) for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The methods are A$$(\alpha)$$-stable for the step length $$k\leq 7$$.

### MSC:

 65L05 Numerical methods for initial value problems involving ordinary differential equations 65L04 Numerical methods for stiff equations 34A34 Nonlinear ordinary differential equations and systems 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations

RODAS
Full Text:

### References:

 [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 [2] Butcher, J. C.: The Numerical Analysis of Ordinary Differential Equation: Runge Kutta and General Linear Methods. Wiley, Chichester, 1987. · Zbl 0616.65072 [3] Butcher, J. C.: Some new hybrid methods for IVPs. Cash, J.R., Gladwell, I. (eds) Computational Ordinary Differential Equations Clarendon Press, Oxford, 1992, 29-46. · Zbl 0769.65037 [4] Butcher, J. C.: High Order A-stable Numerical Methods for Stiff Problems. Journal of Scientific Computing 25 (2005), 51-66. · Zbl 1203.65106 [5] Butcher, J. C.: Forty-five years of A-stability. Numerical Analysis and Applied Mathematics: International Conference on Numerical Analysis and Applied Mathematics 2008. AIP Conference Proceedings 1048 (2008). · Zbl 1191.65109 [6] Butcher, J. C.: Numerical Methods for Ordinary Differential Equations. sec. edi., Wiley, Chichester, 2008. · Zbl 1167.65041 [7] Butcher, J. C.: General linear methods for ordinary differential equations. Mathematics and Computers in Simulation 79 (2009), 1834-1845. · Zbl 1159.65333 [8] Butcher, J. C.: Trees and numerical methods for ordinary differential equations. Numerical Algorithms 53 (2010), 153-170. · Zbl 1184.65072 [9] Butcher, J. C., Hojjati, G.: Second derivative methods with RK stability. Numer. Algorithms 40 (2005), 415-429. · Zbl 1084.65069 [10] Butcher, J. C., Rattenbury, N.: ARK Methods for Stiff Problems. Appl. Numer. Math. 53 (2005), 165-181. · Zbl 1070.65059 [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. · Zbl 0971.65073 [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. [13] Enright, W. H.: Second derivative multistep methods for stiff ODEs. SIAM J. Num. Anal. 11 (1974), 321-331. · Zbl 0249.65055 [14] Enright, W. H.: Continuous numerical methods for ODEs with defect control. J. Comput. Appl. Math. 125 (2000), 159-170. · Zbl 0982.65078 [15] Enright, W. H., Hull, T. E., Linberg, B.: Comparing numerical Methods for Stiff of ODEs systems. BIT 15 (1975), 1-48. · Zbl 0301.65040 [16] Fatunla, S. O.: Numerical Methods for Initial Value Problems in ODEs. Academic Press, New York, 1978. [17] Forrington, C. V. D.: Extensions of the predictor-corrector method for the solution of systems of ODEs. Comput. J. 4 (1961), 80-84. · Zbl 0097.11703 [18] Gear, C. W.: The automatic integration of stiff ODEs. Morrell, A.J.H. (ed.) Information processing 68: Proc. IFIP Congress, Edinurgh, 1968 Nort-Holland, Amsterdam, 1968, 187-193. [19] Gear, C. W.: The automatic integration of ODEs. Comm. ACM 14 (1971), 176-179. · Zbl 0217.21701 [20] Gragg, W. B., Stetter, H. J.: Generalized multistep predictor corrector methods. J. Assoc. Comput. Mach. 11 (1964), 188-209. · Zbl 0168.13803 [21] Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, 1996. · Zbl 0859.65067 [22] Higham, J. D., Higham, J. N.: Matlab Guide. SIAM, Philadelphia, 2000. · Zbl 0953.68642 [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. [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 [25] Lambert, J. D.: Numerical Methods for Ordinary Differential Systems. The Initial Value Problems. Wiley, Chichester, 1991. · Zbl 0745.65049 [26] Lambert, J. D.: Computational Methods for Ordinary Differential Systems. The Initial Value Problems. Wiley, Chichester, 1973. [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. [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. [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. · Zbl 1291.65202 [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. · Zbl 1244.65098 [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. · Zbl 1005.65072 [32] Widlund, O.: A note on unconditionally stable linear multistep methods. BIT 7 (1967), 65-70. · Zbl 0178.18502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.