Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

A Cell-Centered Nonlinear Finite Volume Scheme Preserving Fully Positivity for Diffusion Equation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In the construction of existing nonlinear cell-centered finite volume schemes with monotonicity, it is required to assume that values of auxiliary unknowns are nonnegative. However, this assumption is not always satisfied, especially when accurate reconstruction of auxiliary unknowns is concerned on distorted meshes. In this paper we propose a new method to deal with this issue by introducing both edge unknowns and vertex unknowns as auxiliary unknowns. Edge unknowns are approximated by a convex combination of cell-centered unknowns and vertex unknowns by using the continuity of flux on cell edge. Vertex unknowns are approximated by a convex combination of cell-centered unknowns and edge unknowns. Our new method can assure that these weighted coefficients are nonnegative and the sum of these coefficients in each convex combination is one. The resulting scheme is a nonlinear monotone scheme with nonlinear coefficients depending on both edge unknowns and vertex unknowns, and a linear cell-centered finite volume scheme is formed at each nonlinear iteration by using the Picard linearized method. Numerical results show that our monotone scheme based on the new method of eliminating auxiliary unknowns is more accurate and robust than some existing monotone schemes.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

References

  1. Friis, H.A., Edwards, M.G.: A family of MPFA finite-volume schemes with full pressure support for the general tensor pressure equation on cell-centered triangular grids. J. Comput. Phys. 230, 205–231 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Lipnikov, K., Manzini, G., Svyatskiy, D.: Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems. J. Comput. Phys. 230, 2620–2642 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lu, C., Huang, W., Qiu, J.: Maximum principle in linear finite element approximations of anisotropic diffusion–convection–reaction problems. Numer. Math. 127, 515–537 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Nordbotten, J.M., Aavatsmark, I., Eigestad, G.T.: Monotonicity of control volume methods. Numer. Math. 106, 255–288 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Angelini, O., Chavant, C., Chenier, E., Eymard, R.: A finite volume scheme for diffusion problems on general meshes applying monotony constraints. SIAM J. Numer. Anal. 47, 4193–4213 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burdakov, O., Kapyrin, I., Vassilevski, Y.: Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions. J. Comput. Phys. 231, 3126–3142 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Huang, W.: Discrete maximum principle and a Delaunay-type mesh condition for linear finite element approximations of two-dimensional anisotropic diffusion problems. Numer. Math. Theory Methods Appl. 4, 319–334 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Liska, R., Shashkov, M.: Enforcing the discrete maximum principle for linear finite element solutions of second-order elliptic problems. Commun. Comput. Phys. 3, 852–877 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Lu, C., Huang, W., Van Vleck, E.S.: The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations. J. Comput. Phys. 242, 24–36 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, J., Zhang, R.: Maximum principles for P\(_1\)-conforming finite element approximations of quasi-linear second order elliptic equations. SIAM J. Numer. Anal. 50, 626–642 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wang, S., Yuan, G., Li, Y., Sheng, Z.: Discrete maximum principle based on repair technique for diamond type scheme of diffusion problems. Int. J. Numer. Meth. Fluids 70, 1188–1205 (2012)

    Article  MathSciNet  Google Scholar 

  12. Burman, E., Ern, A.: Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes. C. R. Acad. Sci. Paris Ser. I 338, 641–646 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Le Potier, C.: Finite volume monotone scheme for highly anisotropic diffusion operators on unstructured triangular meshes. C. R. Acad. Sci. Paris Ser. I 341, 787–792 (2005)

    Article  MATH  Google Scholar 

  15. Yuan, G., Sheng, Z.: Monotone finite volume schemes for diffusion equations on polygonal meshes. J. Comput. Phys. 227, 6288–6312 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sheng, Z., Yuan, G.: A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes. SIAM J. Sci. Comput. 30, 1341–1361 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sheng, Z., Yuan, G.: An improved monotone finite volume scheme for diffusion equation on polygonal meshes. J. Comput. Phys. 231, 3739–3754 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Queiroz, L.E.S., Souza, M.R.A., Contreras, F.R.L., Lyra, P.R.M., de Carvalho, D.K.E.: On the accuracy of a nonlinear finite volume method for the solution of diffusion problems using different interpolations strategies. Int. J. Numer. Meth. Fluids 74, 270–291 (2014)

    Article  MathSciNet  Google Scholar 

  19. Gao, Z., Wu, J.: A second-order positivity-preserving finite volume scheme for diffusion equaitons on general meshes. SIAM J. Sci. Comput. 37, A420–A438 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lipnikov, K., Svyatskiy, D., Vassilevski, Y.: Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes. J. Comput. Phys. 228, 703–716 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lipnikov, K., Svyatskiy, D., Vassilevski, Y.: A monotone finite volume method for advection-diffusion equations on unstructured polygonal meshes. J. Comput. Phys. 229, 4017–4032 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sheng, Z., Yue, J., Yuan, G.: Monotone finite volume schemes of nonequilibrium radiation diffusion equations on distorted meshes. SIAM J. Sci. Comput. 31, 2915–2934 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bertolazzi, E., Manzini, G.: A second-order maximum principle preserving volume method for steady convection–diffusion problems. SIAM J. Numer. Anal. 43, 2172–2199 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Cances, C., Cathala, M., Le Potier, C.: Monotone corrections for generic cell-centered finite volume approximations of anisotropic diffusion equations. Numer. Math. 125, 387–417 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Droniou, J., Potier, C.L.: Construction and convergence study of schemes preserving the elliptic local maximum principle. SIAM J. Numer. Anal. 49, 459–490 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Le Potier, C.: A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators. Int. J. Finite Vol. 6, 1–20 (2009)

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Droniou, J.: Finite volume schemes for diffusion equations: introduction to and review of modern methods. Math. Models Methods Appl. Sci. (M3AS) 24, 1575–1619 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Breil, J., Maire, P.-H.: A cell-centered diffusion scheme on two-dimensional unstructured meshes. J. Comput. Phys. 224, 785–823 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Herbin, R., Hubert, F.: Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In: R. Eymard, J.-M. Herard (Eds.), Finite Volumes for Complex Applications V - Problems and Perspectives, pp. 659–692. Wiley Press, London (2008)

Download references

Acknowledgments

This work is partially supported by the National Nature Science Foundation of China (11571047, 11571048), and the Foundation of CEAP (2015B0202042).

Author information

Authors and Affiliations

  1. Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, China

    Zhiqiang Sheng & Guangwei Yuan

Authors
  1. Zhiqiang Sheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guangwei Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhiqiang Sheng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sheng, Z., Yuan, G. A Cell-Centered Nonlinear Finite Volume Scheme Preserving Fully Positivity for Diffusion Equation. J Sci Comput 68, 521–545 (2016). https://doi.org/10.1007/s10915-015-0148-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0148-7

Keywords

Search

Navigation