Advertisement

A Regularized Finite Volume Numerical Method for the Extended Porous Medium Equation Relevant to Moisture Dynamics with Evaporation in Non-woven Fibrous Sheets

  • Hidekazu YoshiokaEmail author
  • Dimetre Triadis
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 603)

Abstract

The extended porous medium equation (PME) is a degenerate nonlinear diffusion equation that effectively describes moisture dynamics with evaporation in non-woven fibrous sheets. We propose a new finite volume numerical model of the extended PME incorporating regularization of nonlinear degenerate terms, and apply it to test cases for verification of accuracy, stability, and versatility. One of the test cases considered is a new exact steady solution of the extended PME. We also examine a differential equation-based adaptive re-meshing technique for resolving sharp transitions of solution profiles that may be optionally incorporated into the procedure above. The computational results demonstrate satisfactory accuracy of the proposed numerical model, with reasonable reproduction of complicated moisture dynamics involving sharp transitions and divorce of supports.

Keywords

Moisture dynamics Evaporation Non-woven fibrous sheet Extended porous medium equation Dual-finite volume method 

Notes

Acknowledgements

We thank Dr. Ichiro Kita and Dr. Kotaro Fukada in Faculty of Life and Environmental Science, Shimane University, Japan for providing helpful comments and suggestions on this article.

References

  1. 1.
    Yoshioka, H., Ito, Y., Kita, I., Fukada, K.: A stable finite volume method for extended porous medium equations and its application to identifying physical properties of a thin non-woven fibrous sheet. In: Proceedings of JSST2015, pp. 398–401 (2015)Google Scholar
  2. 2.
    Landeryou, M., Eames, I., Cottenden, A.: Infiltration into inclined fibrous sheets. J. Fluid Mech. 529, 173–193 (2005)CrossRefzbMATHGoogle Scholar
  3. 3.
    Wilhelmsson, H.: Simultaneous diffusion and reaction processes in plasma dynamics. Phys. Rev. A 38, 1482–1489 (1988)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Haerns, J., Van Gorder, R.A.: Classical implicit travelling wave solutions for a quasilinear convection-diffusion equation. New Astron. 17, 705–710 (2012)CrossRefGoogle Scholar
  5. 5.
    Broadbridge, P., White, I.: Constant rate rainfall infiltration: a versatile nonlinear model. Analytic solution. Water Resour. Res. 24, 145–154 (1988)CrossRefGoogle Scholar
  6. 6.
    Lockington, D.A., Parlange, J.Y., Lenkopane, M.: Capillary absorption in porous sheets and surfaces subject to evaporation. Transport Porous Med. 68, 29–36 (2007)CrossRefGoogle Scholar
  7. 7.
    Brooks, R.H., Corey, A.T.: Hydraulic properties of porous media. Colorado State University, Hydrology Papers, Fort Collins, Colorado (1964)Google Scholar
  8. 8.
    Stewart, J.M., Broadbridge, P.: Calculation of humidity during evaporation from soil. Adv. Water Resour. 22, 495–505 (1999)CrossRefGoogle Scholar
  9. 9.
    Pop, I.S., Radu, F., Knabner, P.: Mixed finite elements for the Richards’ equation: linearization procedure. J. Comput. Appl. Math. 168, 365–373 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Yoshioka, H.: On dual-finite volume methods for extended porous medium equations, arXiv preprint. arXiv:1507.05281 (2015)
  11. 11.
    Yoshioka, H., Unami, K.: A cell-vertex finite volume scheme for solute transport equations in open channel networks. Prob. Eng. Mech. 31, 30–38 (2013)CrossRefGoogle Scholar
  12. 12.
    Li, Y., Lee, G., Jeong, D., Kim, J.: An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation. Comput. Math Appl. 60, 1591–1606 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Huang, W., Russel, R.D.: Adaptive Moving Mesh Methods, pp. 27–133. Springer, Heidelberg (2011)Google Scholar
  14. 14.
    Yoshioka, H., Unami, K., Fujihara, M.: A Petrov-Galerkin finite element scheme for 1-D tome-independent Hamilton-Jacobi-Bellman equations. J. JSCE. Ser. A2, 71 (in press)Google Scholar
  15. 15.
    Yaegashi, Y., Yoshioka, H., Unami, K., Fujihara, M.: An adaptive finite volume scheme for Kolmogorov’s forward equations in 1-D unbounded domains. J. JSCE. Ser. A2, 71 (in press)Google Scholar
  16. 16.
    Li, H., Farthing, M.W., Dawson, C.N., Miller, C.T.: Local discontinuous Galerkin approximations to Richards’ equation. Adv. Water Resour. 30, 555–575 (2007)CrossRefGoogle Scholar
  17. 17.
    Zhang, Q., Wu, Z.L.: Numerical simulation for porous medium equation by local discontinuous Galerkin finite element method. J. Sci. Comput. 38, 127–148 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Aronson, D.G., Caffarelli, L.A.: The initial trace of a solution of the porous medium equation. Trans. Am. Math. Soc. 280, 351–366 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fila, M., Vázquez, J.L., Winkler, M., Yanagida, E.: Rate of convergence to Barenblatt profiles for the fast diffusion equation. Arch. Ration. Mech. An. 204, 599–625 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rosenau, P., Kamin, S.: Thermal waves in an absorbing and convecting medium. Physica D 8, 273–283 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nakaki, T., Tomoeda, K.: A finite difference scheme for some nonlinear diffusion equations in an absorbing medium: support splitting phenomena. SIAM J. Numer. Anal. 40, 945–954 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tomoeda, K.: Numerically repeated support splitting and merging phenomena in a porous media equation with strong absorption. J. Math-for-Ind. 3, 61–68 (2011)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Tomoeda, K.: Numerical and mathematical approach to support splitting and merging phenomena in the behaviour of non-stationary seepage. Theor. Appl. Mech. Jpn. 63, 15–23 (2015)Google Scholar
  24. 24.
    Hayek, M.: Water pulse migration through semi-infinite vertical unsaturated porous column with special relative-permeability functions: exact solutions. J. Hydrol. 517, 668–676 (2014)CrossRefGoogle Scholar
  25. 25.
    Vazquez, J.L.: The Porous Medium Equation. Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  26. 26.
    Atkinson, K., Han, W.: Theoretical Numerical Analysis, pp. 449–451. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  27. 27.
    Bian, S., Liu, J.G.: Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent m > 0. Commun. Math. Phys. 323, 1017–1070 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Naldi, G., Cavalli, F., Perugia, I.: Discontinuous Galerkin approximation of porous Fisher-Kolmogorov equations. Commun. Appl. Indust. Math. 4 (2013). doi: 10.1685/journal.caim.446

Copyright information

© Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Faculty of Life and Environmental ScienceShimane UniversityMatsueJapan
  2. 2.Institute of Mathematics for Industry, Kyushu University - Australia BranchLa Trobe UniversityMelbourneAustralia

Personalised recommendations