Initial value problems for systems of ordinary differential equations (ODEs) are solved numerically by using a combination of (a) the θ-method, (b) the sequential splitting procedure and (c) Richardson Extrapolation. Stability results for the combined numerical method are proved. It is shown, by using numerical experiments, that if the combined numerical method is stable, then it behaves as a second-order method.
@article{bwmeta1.element.doi-10_2478_s11533-011-0099-7,
author = {Zahari Zlatev and Istv\'an Farag\'o and \'Agnes Havasi},
title = {Richardson Extrapolation combined with the sequential splitting procedure and the $\theta$-method},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {159-172},
zbl = {1250.65097},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0099-7}
}
Zahari Zlatev; István Faragó; Ágnes Havasi. Richardson Extrapolation combined with the sequential splitting procedure and the θ-method. Open Mathematics, Tome 10 (2012) pp. 159-172. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0099-7/
[1] Anderson E., Bai Z., Bischof C., Demmel J., Dongarra J., Du Croz J., Greenbaum A., Hammarling S., McKenney A., Ostrouchov S., Sorensen D., LAPACK: Users’ Guide, SIAM, Philadelphia, 1992 | Zbl 0843.65018
[2] Burrage K., Parallel and Sequential Methods for Ordinary Differential Equations, Numer. Math. Sci. Comput., Oxford University Press, New York, 1992
[3] Butcher J.C., Numerical Methods for Ordinary Differential Equations, 2nd ed., John Wiley & Sons, Chichester, 2008 http://dx.doi.org/10.1002/9780470753767 | Zbl 1167.65041
[4] Chin S.A., Geiser J., Multi-product operator splitting as a general method of solving autonomous and nonautonomous equations, IMA J. Numer. Anal. (in press), DOI: 10.1093/imanum/drq022 | Zbl 1232.65174
[5] Dahlquist G.G., A special stability problem for linear multistep methods, Nordisk Tidskr. Informationsbehandling (BIT), 1963, 3, 27–43 | Zbl 0123.11703
[6] Ehle B.L., On Pade Approximations to the Exponential Function and A-stable Methods for the Numerical Solution of Initial Value Problems, PhD thesis, University of Waterloo, 1969
[7] Faragó I., Havasi Á., Operator Splittings and their Applications, Math. Res. Dev. Ser., Nova Science Publishers, Hauppauge, 2009
[8] Faragó I., Havasi Á., Zlatev Z., Efficient implementation of stable Richardson extrapolation algorithms, Comput. Math. Appl., 2010, 60(8), 2309–2325 http://dx.doi.org/10.1016/j.camwa.2010.08.025 | Zbl 1205.65014
[9] Faragó I., Thomsen P.G., Zlatev Z., On the additive splitting procedures and their computer realization, Appl. Math. Model., 2008, 32(8), 1552–1569 http://dx.doi.org/10.1016/j.apm.2007.04.017 | Zbl 1176.65065
[10] Geiser J., Tanoglu G., Operator-splitting methods via the Zassenhaus product formula, Appl. Math. Comput., 2011, 217(9), 4557–4575 http://dx.doi.org/10.1016/j.amc.2010.11.007 | Zbl 1209.65099
[11] Hairer E., Wanner G., Solving Ordinary Differential Equations. II, Springer Ser. Comput. Math., 14, Springer, Berlin, 1991 | Zbl 0729.65051
[12] Hundsdorfer W., Verwer J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Ser. Comput. Math., 33, Springer, Berlin, 2003 | Zbl 1030.65100
[13] Lambert J.D., Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, Chichester, 1991 | Zbl 0745.65049
[14] Richardson L.F., The deferred approach to the limit I. Single lattice, Philos. Trans. Roy. Soc. London Ser. A, 1927, 226, 299–349 http://dx.doi.org/10.1098/rsta.1927.0008
[15] Simpson D., Fagerli H., Jonson J.E., Tsyro S., Wind P., Tuovinen J.-P., Transboundary Acidification, Eutrophication and Ground Level Ozone in Europe. I, Unified EMEP Model Description, EMEP/MSC-W Status Report, 1/2003, Norwegian Meteorological Institute, Oslo, 2003
[16] Wilkinson J.H., The Algebraic Eigenvalue Problem, Clarendon Press, Oxford-London, 1965 | Zbl 0258.65037
[17] Zlatev Z., Modified diagonally implicit Runge-Kutta methods, SIAM J. Sci. Statist. Comput., 1981, 2(3), 321–334 http://dx.doi.org/10.1137/0902026 | Zbl 0475.65040
[18] Zlatev Z., Computer Treatment of Large Air Pollution Models, Environmental Science and Technology Library, 2, Kluwer, Dordrecht-Boston-London, 1995 http://dx.doi.org/10.1007/978-94-011-0311-4
[19] Zlatev Z., Dimov I., Computational and Numerical Challenges in Environmental Modelling, Stud. Comput. Math., 13, Elsevier, Amsterdam, 2006 | Zbl 1120.65103
[20] Zlatev Z., Faragó I., Havasi Á., Stability of the Richardson extrapolation applied together with the θ-method, J. Comput. Appl. Math., 2010, 235(2), 507–517 http://dx.doi.org/10.1016/j.cam.2010.05.052 | Zbl 1201.65134