Abstract
The operator splitting method is a widely used technique which is frequently applied to the solution of complex problems. However, its application is not enough to the practical solution of the problems. The split sub-problems still require some numerical method. In this paper we give a unified investigation of the operator splitting and the numerical discretization. Moreover, we consider the interaction of the operator splitting method and the applied numerical methods to the solution of the different sub-processes. We show that many well-known fully-discretized numerical schemes to solving the Cauchy problems can be obtained in this manner. We investigate the convergence of these methods, too.
Supported by Hungarian National Research Founds (OTKA) under grant N. T043765 and NATO Collaborative Linkage Grant N. 980505.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Csomós, P., Faragó, I., Havasi, Á.: Weighted sequential splittings and their analysis. Comput. Math. Appl. 50, 1017–1031 (2005)
Dekker, K., Verwer, J.G.: Stability of Runge-Kutta methods for stiff nonlinear differential equations. North-Holland, Amsterdam (1984)
Dimitriu, G.: Parameter identification in a two-dimensional parabolic equation using an ADI based solver. In: Margenov, S., Waśniewski, J., Yalamov, P. (eds.) LSSC 2001. LNCS, vol. 2179, pp. 479–486. Springer, Heidelberg (2001)
Faragó, I.: Splitting methods for abstract Cauchy problems. In: Li, Z., Vulkov, L.G., Waśniewski, J. (eds.) NAA 2004. LNCS, vol. 3401, pp. 35–45. Springer, Heidelberg (2005)
Horváth, R.: Uniform Treatment of the Numerical Time-Integration of the Maxwell Equations. Lecture Notes in Computational Science and Engineering, pp. 231–239. Springer, Berlin (2003)
Hundsdorfer, W., Verwer, J.: Numerical solution of time-dependent advectiondiffusion- reaction equations. Springer, Berlin (2003)
Kellogg, R.B.: Another alternating-direction-implicit method. J. Soc. Indust. Appl. Math. 11, 976–979 (1963)
Marchuk, G.: Some applicatons of splitting-up methods to the solution of problems in mathematical physics. Aplikace Matematiky 1, 103–132 (1968)
Marchuk, G.: Splitting and alternating direction methods. North Holland, Amsterdam (1990)
Strang, G.: Accurate partial difference methods I: Linear Cauchy problems. Archive for Rational Mechanics and Analysis 12, 392–402 (1963)
Strang, G.: On the construction and comparison of difference schemes. SIAM J. Num. Anal. 5, 506–517 (1968)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Faragó, I. (2006). Operator Splittings and Numerical Methods. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2005. Lecture Notes in Computer Science, vol 3743. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11666806_39
Download citation
DOI: https://doi.org/10.1007/11666806_39
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-31994-8
Online ISBN: 978-3-540-31995-5
eBook Packages: Computer ScienceComputer Science (R0)