Summary
In this paper we discuss the adaptive solution of parabolic problems. Concentrating on adaptivity in time we describe a time-stepping strategy which is based on extrapolation techniques. The mathematical background in terms of asymptotic expansions and error estimators is presented. The resulting algorithm offers two different sources of parallelism and benefits mainly from basic properties of the time-parallel multigrid method. Numerical results are reported reflecting the sensitivity of the time-stepping strategy.
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
Winfried Auzinger. On the Error Structure of the Implizit Euler Scheme Applied to Stiff Systems of Differential Equations, Computing, Vol. 43, pp. 115–131, 1989.
Winfried Auzinger. On Error Structures and Extrapolation for Stiff Systems with Application in the Method of Lines, Computing, Vol. 44, pp. 331–356, 1990.
Winfried Auzinger, R. Frank and F. Macsek. Asymptotic Error Expansion for Stiff Equations: The implicite Euler Scheme, SIAM J. of Num. Anal., Vol. 27, No. 1, pp. 66–104, 1990.
Peter Bastian, Jens Burmeister, and Graham Horton. Implementation of a parallel multi-grid method for parabolic partial differential equations.In Wolfgang Hackbusch, Editor, Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM-Seminar, Kiel, January 19–21, 1990, pages 18–27, Notes on Numerical Fluid Mechanics, Volume 31, Vieweg-Verlag, Braunschweig, 1991.
Jens Burmeister. Paralleles Lösen diskreter parabolischer Probleme mit Mehrgittertechniken.Diplomarbeit, Universität Kiel, 1985.
Jens Burmeister and Graham Horton. Time-parallel multigrid solution of the Navier-Stokes equations.In Wolgang Hackbusch and Ulrich Trottenberg, editors, Multigrid Methods III. Proceedings of the 3rd European Conference on Multigrid Methods, Bonn, October 1–4, 1990, pages 155–166, International Series of Numerical Mathematics, Vol. 98, Birkhäuser Verlag, Basel, 1991.
Jens Burmeister und Rainer Paul. Schrittweitensteuerung für ein zeitparalleles Mehrgitterverfahren.Bericht Nr. 9314, Institut für Informatik und Praktische Mathematik, Universität Kiel, Oktober 1993.
Wolfgang Hackbusch. Parabolic multi-grid methods.In R. Glowinski and J.-R. Lions, editors, Computing methods in applied sciences and engineering, VI. Proceedings of the 6th International Symposium on Computing Methods in Applied Sciences and Engineering. Versaille, France, December 12–16, 1983, North Holland, 1984.
Wolfgang Hackbusch. Multi-Grid Methods and Applications, Volume 4 of Springer Series in Computational Mathematics.Springer-Verlag, Berlin, Heidelberg, 1985.
Graham Horton. Ein zeitparalleles Lösungsverfahren für die Navier-Stokes- Gleichungen.PhD thesis, Technische Fakultät der Universität Erlangen-Nürnberg, Erlangen, 1991.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden
About this chapter
Cite this chapter
Burmeister, J., Paul, R. (1995). Time-Adaptive Solution of Discrete Parabolic Problems with Time-Parallel Multigrid Methods. In: Wagner, S. (eds) Computational Fluid Dynamics on Parallel Systems. Notes on Numerical Fluid Mechanics (NNFM), vol 50. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-322-89454-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-322-89454-0_5
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-528-07650-4
Online ISBN: 978-3-322-89454-0
eBook Packages: Springer Book Archive