Skip to main content

A Finite Volume Scheme with the Discrete Maximum Principle for Diffusion Equations on Polyhedral Meshes

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 77)

Abstract

We present a cell-centered finite volume (FV) scheme with the compact stencil formed mostly by the closest neighboring cells. The discrete solution satisfies the discrete maximum principle and approximates the exact solution with second-order accuracy. The coefficients in the FV stencil depend on the solution; therefore, the FV scheme is nonlinear. The scheme is applied to the steady state diffusion equation discretized on a general polyhedral mesh.

Keywords

  • Diffusive Flux
  • Collocation Point
  • Numerical Flux
  • Finite Volume Scheme
  • Discrete Maximum Principle

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-05684-5_18
  • Chapter length: 9 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   129.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-05684-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   169.00
Price excludes VAT (USA)
Hardcover Book
USD   179.99
Price excludes VAT (USA)
Fig. 1

References

  1. FVCA6 3D Benchmark. http://www.latp.univ-mrs.fr/latp_numerique/?q=node/4

  2. Agelas, L., Eymard, R., Herbin, R.: A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media. C. R. Acad. Sci. Paris Ser. I. 347, 673–676 (2009)

    Google Scholar 

  3. Chernyshenko, A.: Generation of adaptive polyhedral meshes and numerical solution of 2nd order elliptic equations in 3D domains and on surfaces. Ph.D. thesis, INM RAS, Moscow (2013).

    Google Scholar 

  4. Danilov, A., Vassilevski, Y.: A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes. Russ. J. Numer. Anal. Math. Model. 24(3), 207–227 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Droniou, J.: Finite volume schemes for diffusion equations: introduction to and review of modern methods. Math. Models Methods Appl. Sci. (2014). To appear

    Google Scholar 

  6. Gao, Z.M., Wu, J.M.: A small stencil and extremum-preserving scheme for anisotropic diffusion problems on arbitrary 2d and 3d meshes. J. Comp. Phys. 250, 308–331 (2013)

    CrossRef  Google Scholar 

  7. LePotier, C.: Schema volumes finis monotone pour des operateurs de diffusion fortement anisotropes sur des maillages de triangle non structures. C. R. Acad. Sci. Paris Ser. I. 341, 787–792 (2005)

    Google Scholar 

  8. Lipnikov, K., Svyatskiy, D., Shashkov, M., Vassilevski, Y.: Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. J. Comp. Phys. 227, 492–512 (2007)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Lipnikov, K., Svyatskiy, D., Vassilevski, Y.: Minimal stencil finite volume scheme with the discrete maximum principle. Russ. J. Numer. Anal. Math. Model. 27(4), 369–385 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. Lipnikov, K., Svyatskiy, D., Vassilevski, Y.: Anderson acceleration for nonlinear finite volume scheme for advection-diffusion problems. SIAM J. Sci. Comput. 35(2), 1120–1136 (2013)

    CrossRef  MathSciNet  Google Scholar 

  11. Yuan, G., Sheng, Z.: The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes. J. Comp. Phys. 230(7), 2588–2604 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work has been supported in part by RFBR grants 12-01-33084, 14-01-00830, Russian Presidential grant MK-7159.2013.1, Federal target programs of Russian Ministry of Education and Science, ExxonMobil Upstream Research Company, and project “Breakthrough” of Rosatom.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alexey Chernyshenko .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Chernyshenko, A., Vassilevski, Y. (2014). A Finite Volume Scheme with the Discrete Maximum Principle for Diffusion Equations on Polyhedral Meshes. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds) Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects. Springer Proceedings in Mathematics & Statistics, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-319-05684-5_18

Download citation