Skip to main content
SpringerLink
Log in

Additive splitting methods for elliptic-parabolic problems

  • Published:
Annali dell’Università di Ferrara Aims and scope Submit manuscript

Sunto

In questo lavoro vengono proposti tre diversi schemi di decomposizione alle differenze finite per la risoluzione numerica di problemi ellittico-parabolici non lineari in 3D. Negli algoritmi sono incluse strategie adattative front-tracking e time-stepping. La parallelizzazione degli algoritmi è realizzata usando il metodo della decomposizione di domini. Viene impiegata la decomposizione 1D del dominio computazionale per ottenere il bilanciameto ottimale del carico computazionale tra i processori e per minizzare la frequenza della comunicazione dei dati. Durante le computazioni, infine, viene realizzata dinamicamente la ridistribuzione dei domini computazionali.

Abstract

Three finite-difference splitting schemes are proposed for numerical solution of the nonlinear 3D parabolic-elliptic problem. Adaptive front-tracking and time-stepping strategies are included into the algorithms. Parallelization of the algorithms is done using the domain decomposition method. The 1D decomposition of the computational domain is used in order to obtain the optimal computational load balancing among processors and to minimize the frequency of data communications. A redistribution of the computational domain among processors is done dynamically during computations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. H. W. Alt—S. Luckhaus,Quasilinear elliptic-parabolic differential equations, Math. Z.,183 (1983), pp. 311–341.

    Article  MATH  MathSciNet  Google Scholar 

  2. Z. Chen—R. Ewing,Fully discrete finite element analysis of multiphase flow in groundwater hydrology, SIAM J. Numer. Anal.,34 (1997), pp. 2228–2253.

    Article  MATH  MathSciNet  Google Scholar 

  3. X. Chen—A. Friedman—T. Kimura,Nonstationary filtrations in partially saturated porous media. Europ. J. App. Math.,5 (1994), pp. 405–429.

    MATH  MathSciNet  Google Scholar 

  4. R. Čiegis—A. Zemitis,Numerical algorithms for simulation of the liquid transport in multilayered fleeces, Proceedings of the 15th IMACS World Congress, Berlin, August 24–29 1997,2 (1997), pp. 117–122.

    Google Scholar 

  5. R. Čiegis—A. Zemitis The mathematical simulations of the liquid transport in multilayered nonwoven, Report Berichte der Arbeitsgruppe Technomathematik, Kaiserslautern University, No. 184, 1997.

  6. R. Čiegis—R. Čiegis—A. Zemitis,Parallel numerical methods for the ellipticparabolic problem, Progress in Industrial Mathematic at ECMI98, B. G. Teubner, Stuttgart, Leipzig (1999), pp. 206–213.

    Google Scholar 

  7. J. Douglas,Alternating direction method for three space variables, Numer. Math.,4 (1962), pp. 41–63.

    Article  MATH  MathSciNet  Google Scholar 

  8. W. Jäger—J. Kačur,Solution of porous medium systems by linear approximation scheme, Numer. Math.,60 (1991), pp. 407–427.

    Article  MATH  MathSciNet  Google Scholar 

  9. P. A. Forsyth—M. C. Kropinski.Monotonicity considerations for saturated-unsaturated subsurface flow, SIAM J. Sci. Comput.,18 (1997), pp. 1328–1354.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. H. Golub—Ch. F. Van Loan,Matrix Computations, The Johns Hopkins University Press, Aaltmore and London, 1991.

    Google Scholar 

  11. W. Hundsdorfer,A note on stability of the Douglas splitting method, CWI Report NM-R9606, Amsterdam, 1996.

  12. J. Kačur,Solution of some free boundary problems by relaxation schemes, SIAM J. Numer. Anal.,36 (1999), pp. 290–316.

    MathSciNet  Google Scholar 

  13. G. I. Marchuk,Splitting and alternating direction methods, Handbook of Numerical Analysis 1, North-Holland, Amsterdam, pp. 197–462, 1990.

    Google Scholar 

  14. J. Molenaar,Nonlinear multigrid for fully-implicit and high-order accurate simulation of multiphase flow in porous media, CWI Report MAS-R9712, Amsterdam, 1997.

  15. R. H. Nochetto—C. Verdi,Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal.,25 (1988), pp. 784–814.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. LeVeque—J. Oliger,Numerical methods based on additive splittings for hyperbolic partial differential equations, Math. Comput.,40 (1983), pp. 469–497.

    Article  MATH  MathSciNet  Google Scholar 

  17. Ch. Xu—F. Lau,Load Balancing in Parallel Computers: Theory and Practice, Kluwer Academic Publishers, 1997.

Download references

Author information

Author notes

  1. Raimondas Čiegis

    Present address: Institute of Mathematics and Informatics, Akademijos 4, 2600, Vilnius, Lithuania

Authors and Affiliations

  1. ITWM University of Kaiserslautern, Ervin-Schroedinger Str., D-67663, Kaiserslautern, Germany

    Areti Papastavrou & Aivars Zemitis

Authors
  1. Raimondas Čiegis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Areti Papastavrou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aivars Zemitis
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Čiegis, R., Papastavrou, A. & Zemitis, A. Additive splitting methods for elliptic-parabolic problems. Ann. Univ. Ferrara 46, 291–306 (2000). https://doi.org/10.1007/BF02837304

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02837304

Keywords

Search

Navigation