Skip to main content
SpringerLink
Log in

Finite volume ADI scheme for hybrid dimension heat conduction problems set in a cross-shaped domain

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

Abstract

In this paper, we construct an alternating direction implicit (ADI) type finite volume numerical scheme to solve a nonclassical nonstationary heat conduction problem set in a 2D cross-shaped domain. We reduce the original model to a hybrid dimension model in a large part of the domain. We define special conjugation conditions between 2D and 1D parts. We apply the finite volume method to approximate spatial differential operators and use ADI splitting for time integration. The ADI scheme is unconditionally stable, and under a mix of Dirichlet and Neumann boundary conditions, the approximation error is of second order in space and time. The results of computational experiments confirm the theoretical error analysis. We compare visual representations and computational times for various sizes of reduced dimension zones.

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

Similar content being viewed by others

References

  1. H. Amar and D. Givoli, Mixed-dimensional modeling of time-dependent wave problems using the Panasenko construction, J. Theor. Comput. Acoust., 26(03), 2018.

  2. H. Amar and D. Givoli, Mixed-dimensional coupling for time-dependent wave problems using the Nitsche method, Comput. Methods Appl. Mech. Eng., 349:213–250, 2019.

    Article  MathSciNet  Google Scholar 

  3. A. Amosov and G. Panasenko, Partial dimension reduction for the heat equation in a domain containing thin tubes, Math. Methods Appl. Sci., 41(18):9529–9545, 2018.

    Article  MathSciNet  Google Scholar 

  4. U.M. Ascher, Numerical Methods for Evolutionary Differential Equations, SIAM, Philadelphia, 2008.

    Book  Google Scholar 

  5. C. Bertoglio, C. Conca, D. Nolte, G. Panasenko, and K. Pileckas, Junction of models of different dimension for flows in tube structures by Womersley-type interface conditions, SIAM J. Appl. Math., 79(3):959–985, 2019.

    Article  MathSciNet  Google Scholar 

  6. E. Canon, F. Chardard, G. Panasenko, and O. Štikonienė, Numerical solution of the viscous flows in a network of thin tubes: Equations on the graph, J. Comput. Phys., 435:110262, 2021.

    Article  MathSciNet  Google Scholar 

  7. R.C. Cascaval, C. D’Apice, M.P. D’Arienzo, and R. Manzo, Flow optimization in vascular networks, Math. Biosci. Eng., 14(3):607–624, 2017.

    Article  MathSciNet  Google Scholar 

  8. R. Čiegis, G. Panasenko, K. Pileckas, and V. Šumskas, ADI scheme for partially dimension reduced heat conduction models, Comput. Math. Appl., 80(5):1275–1286, 2020.

    Article  MathSciNet  Google Scholar 

  9. W.J. De Lange et al., An operational, multi-scale, multi-model system for consensus-based, integrated water management and policy analysis: The Netherlands hydrological instrument, Environ. Model. Softw., 59:98–108, 2014.

    Article  Google Scholar 

  10. W.H. Hundsdorfer and J.G. Verwer, Stability and convergence of the Peaceman–Rachford ADI method for initialboundary value problems, Math. Comput., 53(187):81–101, 1989.

    Article  Google Scholar 

  11. W.H. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-ReactionEquations, Springer Ser. Comput. Math., Vol. 33, Springer, Berlin, Heidelberg, 2003.

  12. O.S. Ismail and G.T. Adewoye, Analyses and modeling of laminar flow in pipes using numerical approach, J. Softw. Eng. Appl., 5(9):653–658, 2012.

    Article  Google Scholar 

  13. G.I. Marchuk, Splitting and alternating direction methods, Handb. Numer. Anal., 1:197–462, 1990.

    MathSciNet  MATH  Google Scholar 

  14. A. Nachit, G. Panasenko, and A. Zine, Asymptotic partial domain decomposition in thin tube structures: Numerical experiments, Int. J. Multiscale Comput. Eng., 11:407–441, 09 2013.

  15. G. Panasenko, Method of asymptotic partial decomposition of domain, Math. Models Methods Appl. Sci., 8(1):139–156, 1998.

    Article  MathSciNet  Google Scholar 

  16. G. Panasenko, Multi-Scale Modelling for Structures and Composites, Springer, Dordrecht, 2005.

    MATH  Google Scholar 

  17. G. Panasenko and M. Viallon, Finite volume implementation of the method of asymptotic partial domain decomposition for the heat equation on a thin structure, Russ. J. Math. Phys., 22(2):237–263, 2015.

    Article  MathSciNet  Google Scholar 

  18. D.W. Peaceman and H.H. Rachford, The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. App. Math., 3(1):28–41, 1955.

    Article  MathSciNet  Google Scholar 

  19. A. Quarteroni, F. Saleri, and A. Veneziani, Factorization methods for the numerical approximation of Navier–Stokes equations, Comput. Methods. Appl. Mech. Eng., 188(1–3):505–526, 2000.

    Article  MathSciNet  Google Scholar 

  20. B. Rathish Kumar, A. Quateroni, L. Formaggia, and D. Lamponi, On parallel computation of blood flow in human arterial network based on 1-D modelling, Computing, 71:321–351, 2003.

    Article  MathSciNet  Google Scholar 

  21. A.A. Samarskii, Fundamentals of Numerical Reservoir Simulation, Elsevier, Amsterdam, Oxford, New York, 1977.

    Google Scholar 

  22. A.A. Samarskii, The Theory of Difference Schemes, Marcel Dekker, New York, 2001.

    Book  Google Scholar 

  23. M. Sapagovas, G. Kairytė, O. Štikonienė, and A. Štikonas, Alternating direction method for a two-dimensional parabolic equation with a nonlocal boundary condition, Math. Model. Anal., 12(1):131–142, 2007.

    Article  MathSciNet  Google Scholar 

  24. M. Sapagovas, A. Štikonas, and O. Štikonienė, Alternating direction method for the Poisson equation with variable weight coefficients in an integral condition, Differ. Equ., 47:1176–1187, 08 2011.

  25. M. Sapagovas and O. Štikonienė, Alternating-directionmethod for a mildly nonlinear elliptic equation with nonlocal integral conditions, Nonlinear Anal. Model. Control, 16(2):220–230, 2011.

    Article  MathSciNet  Google Scholar 

  26. Y. Shi, P. Lawford, and R. Hose, Review of zero-D and 1-D models of blood flow in the cardiovascular system, BioMed. Eng. Online, 10(33), 2011.

  27. V. Skakauskas, P. Katauskis, and R. Čiegis, Modelling of the NO + CO reaction over inhomogeneous surfaces, J. Math. Chem., 56(9):2626–2642, 2018.

    Article  MathSciNet  Google Scholar 

  28. O. Štikonienė and M. Sapagovas, Numerical investigation of alternating-direction method for Poisson equation with weighted integral conditions, Liet. Mat. Rink., LMD Darbai, 51:385–390, 2010.

  29. M.-C. Viallon, Domain decomposition methods in a geometrical multiscale domain using finite volume schemes, Int. J. Numer. Methods Fluids, 92(5):391–421, 2020.

    Article  MathSciNet  Google Scholar 

  30. V. Volpert, N. Bessonov, A. Sequeira, S. Simakov, and Yu. Vassilevskii, Methods of blood flow modelling, Math. Model. Nat. Phenom., 11(1):1–25, 2016.

    Article  MathSciNet  Google Scholar 

  31. N.N. Yanenko, The Method of Fractional Steps, Springer, Berlin, Heidelberg, 1971.

    Book  Google Scholar 

  32. D. Zaheer, M. Ahsan, M. Ahmad, W. Khan, E.E. Mahmoud, and A. Abdel-Aty, Meshless analysis of nonlocal boundary value problems in anisotropic and inhomogeneous media, Mathematics, 8(11):2045, 2020.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, Vilnius University, Naugarduko str. 24, Vilnius, Lithuania

    Vytenis Šumskas

  2. Vilnius Gediminas Technical University, Saulėtekio av. 11, Vilnius, Lithuania

    Raimondas Čiegis

Authors
  1. Vytenis Šumskas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Raimondas Čiegis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vytenis Šumskas.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Vytenis Šumskas was supported in part by European Social Fund (project No. 09.3.3-LMT-K-712-17-0003) under grant agreement with the Research Council of Lithuania (LMTLT).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Šumskas, V., Čiegis, R. Finite volume ADI scheme for hybrid dimension heat conduction problems set in a cross-shaped domain. Lith Math J 62, 239–258 (2022). https://doi.org/10.1007/s10986-022-09561-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10986-022-09561-0

MSC

Keywords

Search

Navigation