Richardson Extrapolation combined with the sequential splitting procedure and the θ-method
Zahari Zlatev; István Faragó; Ágnes Havasi
Open Mathematics (2012)
- Volume: 10, Issue: 1, page 159-172
- ISSN: 2391-5455
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topZahari Zlatev, István Faragó, and Ágnes Havasi. "Richardson Extrapolation combined with the sequential splitting procedure and the θ-method." Open Mathematics 10.1 (2012): 159-172. <http://eudml.org/doc/269343>.
@article{ZahariZlatev2012,
abstract = {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.},
author = {Zahari Zlatev, István Faragó, Ágnes Havasi},
journal = {Open Mathematics},
keywords = {Initial value problems; Ordinary differential equations; Sequential splitting; Richardson Extrapolation; Stability; Cauchy problem; weighted scheme of quadrature formulas; Richardson extrapolation rule; system; stability},
language = {eng},
number = {1},
pages = {159-172},
title = {Richardson Extrapolation combined with the sequential splitting procedure and the θ-method},
url = {http://eudml.org/doc/269343},
volume = {10},
year = {2012},
}
TY - JOUR
AU - Zahari Zlatev
AU - István Faragó
AU - Ágnes Havasi
TI - Richardson Extrapolation combined with the sequential splitting procedure and the θ-method
JO - Open Mathematics
PY - 2012
VL - 10
IS - 1
SP - 159
EP - 172
AB - 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.
LA - eng
KW - Initial value problems; Ordinary differential equations; Sequential splitting; Richardson Extrapolation; Stability; Cauchy problem; weighted scheme of quadrature formulas; Richardson extrapolation rule; system; stability
UR - http://eudml.org/doc/269343
ER -
References
top- [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 Zbl0843.65018
- [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 Zbl1167.65041
- [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 Zbl1232.65174
- [5] Dahlquist G.G., A special stability problem for linear multistep methods, Nordisk Tidskr. Informationsbehandling (BIT), 1963, 3, 27–43 Zbl0123.11703
- [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 Zbl1205.65014
- [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 Zbl1176.65065
- [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 Zbl1209.65099
- [11] Hairer E., Wanner G., Solving Ordinary Differential Equations. II, Springer Ser. Comput. Math., 14, Springer, Berlin, 1991 Zbl0729.65051
- [12] Hundsdorfer W., Verwer J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Ser. Comput. Math., 33, Springer, Berlin, 2003 Zbl1030.65100
- [13] Lambert J.D., Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, Chichester, 1991 Zbl0745.65049
- [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 Zbl0258.65037
- [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 Zbl0475.65040
- [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 Zbl1120.65103
- [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 Zbl1201.65134
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