×

Comparison of some recent numerical methods for initial-value problems for stiff ordinary differential equations. (English) Zbl 1142.65054

Summary: We consider the combustion equation as one of the candidates from the class of stiff ordinary differential equations. A solution over a length of time that is inversely proportional to \(\delta >0\) (where \(\delta >0\) is a small disturbance of the pre-ignition state) is sought. This problem has a transient at the midpoint of the integration interval. The solution changes from being non-stiff to stiff, and afterwards becomes non-stiff again. We provide its asymptotic and numerical solution obtained via a variety of methods.
Comparisons are made for the numerical results which we obtain with the MATLAB ode solvers (ODE45, ODE15s and ODE23s) and some nonstandard finite difference methods. Results corresponding to standard finite difference method are also presented. Furthermore, the discussion on these approaches along with the others, provides several open problems for new and young researchers.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L12 Finite difference and finite volume methods for ordinary differential equations
80A25 Combustion
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] The Math Works, Inc.: http://www.mathworks.com; The Math Works, Inc.: http://www.mathworks.com
[2] Reiss, E. L., A new asymptotic method for jump phenomena, SIAM J. Appl. Math., 39, 440-455 (1980) · Zbl 0444.34054
[3] Enright, W. H.; Hull, T. E.; Lindberg, B., Comparing numerical methods for stiff systems of ODEs, BIT, 15, 1, 10-48 (1975) · Zbl 0301.65040
[4] Kassoy, D. R., A note on asymptotic methods for jump phenomena, SIAM J. Appl. Math., 42, 926-932 (1982) · Zbl 0518.34050
[5] Brugnano, L.; Trigiante, D., Boundary value methods: The third way between linear multistep and Runge-Kutta methods, Comput. Math. Appl., 36, 10-12, 269-284 (1998) · Zbl 0933.65082
[6] Iavernaro, F.; Mazzia, F., Solving ordinary differential equations by generalized Adams methods: Properties and implementation techniques, Appl. Numer. Math., 28, 107-126 (1998) · Zbl 0926.65076
[7] Iavernaro, F.; Mazzia, F., Block-Boundary Value Methods for the solution of ordinary differential equation, SIAM J. Sci. Comput., 21, 1, 323-339 (1999) · Zbl 0941.65067
[8] Hsiao, C. H., Numerical solution of stiff differential equations via Harr wavelets, Int. J. Comput. Math., 82, 9, 1117-1123 (2005) · Zbl 1075.65098
[9] Jannelli, A.; Fazio, R., Adaptive stiff solvers at low accuracy and complexity, J. Comput. Appl. Math., 191, 246-258 (2006) · Zbl 1089.65061
[10] (Aiken, R. C., Stiff Computation (1985), Oxford University Press: Oxford University Press Oxford, UK) · Zbl 0607.65041
[11] Day, J. D., A minimum configuration L-stable fourth-order non-autonomous Rosenbrock method for stiff differential equations, Commun. Appl. Numer. Methods, 1, 6, 293-297 (2005) · Zbl 0591.65049
[12] Gear, C. W., Numerical Initial Value Problems in Ordinary Differential Equations (1971), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0217.21701
[13] Kaps, P.; Rentrop, P., Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations, Numer. Math., 33, 1, 55-68 (1979) · Zbl 0436.65047
[14] Kaps, P.; Poon, S. W.H.; Bui, T. D., Rosenbrock methods for Stiff ODEs: A comparison of Richardson extrapolation and embedding technique, Computing, 34, 1, 17-40 (1985) · Zbl 0554.65054
[15] Shampine, L. F., Evaluation of a test set for stiff ODE solvers, ACM Trans. Math. Software, 7, 409-420 (1981)
[16] Shampine, L. F.; Baca, L. S., Error estimators for stiff differential equations, J. Comput. Appl. Math., 11, 197-207 (1984) · Zbl 0556.65065
[17] Shampine, L. F., Numerical Solution of Ordinary Differential Equations (1994), Chapman & Hall: Chapman & Hall New York · Zbl 0826.65082
[18] Ueberhuber, C. W., Implementation of defect correction methods for stiff differential equations, Computing, 23, 3, 205-232 (1979) · Zbl 0423.65042
[19] Anguelov, R.; Lubuma, J. M.-S., Nonstandard finite difference method by nonlocal approximation, Math. Comput. Simul., 61, 3-6, 465-475 (2003) · Zbl 1015.65034
[20] (Mickens, R. E., Applications of Nonstandard Finite Difference Schemes (2000), World Scientific: World Scientific Singapore) · Zbl 0989.65101
[21] Mickens, R. E., Nonstandard Finite Difference Models of Differential Equations (1994), World Scientific: World Scientific Singapore · Zbl 0925.70016
[22] Patidar, K. C., On the use of nonstandard finite difference methods. On the use of nonstandard finite difference methods, J. Difference Eq. Appl., 11, 735-758 (2005) · Zbl 1073.65545
[23] Miranker, W. L., Numerical Methods for Stiff Equations and Singular Perturbation Problems (1981), D. Reidel: D. Reidel Holland · Zbl 0461.65061
[24] O’Malley, R. E., Singular Perturbation Methods for Ordinary Differential Equations (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0743.34059
[25] Corless, R. M.; Gonnet, G. H.; Hare, D. E.G.; Jeffrey, D. J.; Knuth, D. E., On the Lambert \(W\) function, Adv. Comput. Math., 5, 329-359 (1996) · Zbl 0863.65008
[26] Dormand, J. R.; Prince, P. J., A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6, 19-26 (1980) · Zbl 0448.65045
[27] Shampine, L. F.; Reichelt, M. W., The MATLAB ODE Suite, SIAM J. Sci. Comput., 18, 1-22 (1997) · Zbl 0868.65040
[28] Shampine, L. F.; Reichelt, M. W.; Kierzenka, J. A., Solving index-1 DAEs in MATLAB and Simulink, SIAM Rev., 41, 538-552 (1999) · Zbl 0935.65082
[29] Gumel, A. B.; Patidar, K. C.; Spiteri, R. J., Asymptotically consistent non-standard finite-difference methods for solving mathematical models arising in population biology, (Mickens, R. E., Advances in the Applications of Nonstandard Finite Difference Schemes (2005), World Scientific: World Scientific Singapore), 385-421 · Zbl 1086.65080
[30] Gottlieb, S.; Shu, C. W., Total variation diminishing Runge-Kutta schemes, Math. Comp., 67, 221, 73-85 (1998) · Zbl 0897.65058
[31] Shu, C. W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 2, 439-471 (1988) · Zbl 0653.65072
[32] Ferracina, L.; Spijker, M. N., Step-size restrictions for the total-variation-diminishing property in general Runge-Kutta methods, SIAM J. Numer. Anal., 42, 3, 1073-1093 (2004) · Zbl 1080.65087
[33] Lubuma, J. M.-S.; Patidar, K. C., Contributions to the theory of non-standard finite difference methods and applications to singular perturbation problems, (Mickens, R. E., Advances in the Applications of Nonstandard Finite Difference Schemes (2005), World Scientific: World Scientific Singapore), 513-560 · Zbl 1085.65070
[34] Patidar, K. C.; Sharma, K. K., \( \varepsilon \)-uniformly convergent nonstandard finite difference methods for singularly perturbed differential difference equations with small delay, Appl. Math. Comput., 175, 1, 864-890 (2006) · Zbl 1096.65071
[35] Patidar, K. C.; Sharma, K. K., Uniformly convergent nonstandard finite difference methods for singularly perturbed differential difference equations with delay and advance, Internat. J. Numer. Methods Engrg., 66, 2, 272-296 (2006) · Zbl 1123.65078
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.