Skip to main content
SpringerLink
Log in

Block-centered finite difference methods for parabolic equation with time-dependent coefficient

  • Original Paper
  • Area 1
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

Two block-centered finite difference schemes are introduced and analyzed to solve parabolic equation with time-dependent diffusion coefficient. One scheme is Euler backward scheme with first order accuracy in time increment while the other is Crank–Nicolson scheme with second order accuracy in time increment. Second-order error estimates in spacial meshsize both for the original unknown and its derivatives in discrete L 2 norms are established on non-uniform rectangular grid. Numerical experiments using the schemes show that the convergence rates are in agreement with the theoretical analysis.

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. Arbogast T., Dawson C.N., Keenan P.T., Wheeler M.F., Yotov I.: Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput. 19, 404–425 (1998)

    Article  MathSciNet  Google Scholar 

  2. Arbogast T., Wheeler M.F., Yotov I.: Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34, 828–852 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aziz K., Settari A.: Petroleum Reservoir Simulation. Applied Science Publishers, London (1979)

    Google Scholar 

  4. Brezzi F., Douglas J. Jr., Marini L.D.: Two families of mixed finite elements for second order elliptic problems. Numerische Mathematik 47((2), 217–235 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dawson C.N., Dupont T.F.: Explicit/implicite, conservative domain decomposition procedure for parabolic problems based on block-centered finite differences. SIAM J. Numer. Anal. 31, 1045–1061 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Peaceman D.W.: Foundaments of Numerical Reservoir Simulation. Elsevier Scientific Publishing Co., Amsterdam (1977)

    Google Scholar 

  7. Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2-nd order elliptic problems. In: Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics, vol. 606, Ch. 19, pp. 292–315. Springer, Berlin (1977)

  8. Rui H., Pan H.: A block-centered finite difference method for the Darcy–Forchheimer model. SIAM J Numer. Anal. 50, 2612–2631 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Russell, T.F., Wheeler, M.F.: Finite element and finite difference methods for continuous flows in porous media. In: The Mathematics of Reservoir Simulation. The Society of Industrial and Applied Mathematics, Philadelphia (1983)

  10. Weiser A., Wheeler M.F.: On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25, 351–375 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wheeler, M.F., Yotov, I.: A cell-centered finite difference method on quadrilaterals. In: Compatible Spatial discretizations. The IMA Volumes in Mathematics and its Applications, vol. 142, pp. 1045–1061. Springer, New York (2006)

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Hongxing Rui

  2. School of Information Science and Engineering, Shandong Agricultural University, Taian, 271018, China

    Hao Pan

Authors
  1. Hongxing Rui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hao Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongxing Rui.

Additional information

The work is supported by the National Natural Science Foundation of China Grant No. 91330106, 11171190 and the Research Fund for Doctoral Program of High Education by Station Education Ministry of China No. 20120131110003.

About this article

Cite this article

Rui, H., Pan, H. Block-centered finite difference methods for parabolic equation with time-dependent coefficient. Japan J. Indust. Appl. Math. 30, 681–699 (2013). https://doi.org/10.1007/s13160-013-0114-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-013-0114-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation