Skip to main content
SpringerLink
Log in

Fitted Finite Volume Method for Unsaturated Flow Parabolic Problems with Space Degeneration

  • Conference paper
  • First Online:
Large-Scale Scientific Computing (LSSC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13127))

Included in the following conference series:

Abstract

In the present work, we discuss a question of correct boundary conditions and adequate approximation of parabolic problems with space degeneration in porous media. To the Richards equation, as a typical problem, we apply a time discretization, linearize the obtained nonlinear problem and introduce correct boundary conditions. Then, we develop fitted finite volume method to get the space discretization of the model problem. A graded space mesh is also deduced. We illustrate experimentally that the proposed method is efficient in the case of degenerate permeability.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Arbogast, T., Taicher, A.L.: A linear degenerate elliptic equation arising from two-phase mixtures. SIAM J. Numer. Anal. 54(5), 3105–3122 (2016)

    Article  MathSciNet  Google Scholar 

  2. Bellman, R., Kalaba, R.: Quasilinearization and Nonlinear Boundary-Value Problems. Elsevier Publishing Company, New York (1965)

    MATH  Google Scholar 

  3. Castro, H., Wang, H.: A singular Sturm-Liouville equation under homogeneous boundary conditions. J. Funct. Anal. 261, 1542–1590 (2011)

    Article  MathSciNet  Google Scholar 

  4. Casulli, V., Zanolli, P.: A nested Newton-type algorithm for finite volume methods solving Richards’ equation in mixed form. SIAM J. Sci. Comput. 32, 2255–2273 (2010)

    Article  MathSciNet  Google Scholar 

  5. Celia, M., Boulout, F., Zarba, R.L.: A general mass-conservative numerical solution for the unsaturated flow equation. Water Resour. Res. 26(7), 1483–1496 (1990)

    Article  Google Scholar 

  6. Chernogorova, T., Koleva, M.N., Vulkov, L.G.: Exponential finite difference scheme for transport equations with discontinuous coefficients in porous media. Appl. Math. Comput. 392(1), 125691 (2021)

    MathSciNet  MATH  Google Scholar 

  7. Dostert, P., Efendiev, Y., Mohanty, B.: Efficient uncertainty quantification techniques in inverse problems for Richards’ equation using coarse-scale simulation models. Adv. Water Resour. 32, 329–339 (2009)

    Article  Google Scholar 

  8. Farthing, M.W., Ogden, F.L.: Numerical solution of Richards’ equation: a review of advances and challenges. Soil Sci. Soc. Amer. J. 81(6), 1257–1269 (2017)

    Article  Google Scholar 

  9. Gardner, W.R.: Some steady-state solutions of the unsaturated moistureflow equation with application to evaporation from a water table. Soil Sci. 85(4), 228–232 (1958)

    Article  Google Scholar 

  10. van Genuchten, M.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44, 892–898 (1980)

    Article  Google Scholar 

  11. Huang, F., Luo, X., Liu, W.: Stability analysis of hydrodynamic pressure landslides with different permeability coefficients affected by reservoir water level fluctuations and rainstorms. Water 9(7), 450 (2017)

    Article  Google Scholar 

  12. Koleva, M.N., Vulkov, L.G.: Weighted time-semidiscretization Quasilinearization method for solving Rihards’ equation. In: Lirkov, I., Margenov, S. (eds.) LSSC 2019. LNCS, vol. 11958, pp. 123–130. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-41032-2_13

    Chapter  Google Scholar 

  13. Ku, C.Y., Liu, C.Y., Xiao, J.E., Yeih, W.: Transient modeling of flow in unsaturated soils using a novel collocation meshless method. Water 9(12), 954 (2017)

    Article  Google Scholar 

  14. Misiats, O., Lipnikov, K.: Second-order accurate finite volume scheme for Richards’ equation. J. Comput. Phys. 239, 125–137 (2013)

    Article  MathSciNet  Google Scholar 

  15. Mitra, K., Pop, I.S.: A modified L-scheme to solve nonlinear diffusion problems. Comput. Math. Appl. 77(6), 1722–1738 (2019)

    Article  MathSciNet  Google Scholar 

  16. Richards, L.A.: Capillary conduction of liquids through porous mediums. Physics 1(5), 318–333 (1931)

    Article  Google Scholar 

  17. Sinai, G., Dirksen, C.: Experimental evidence of lateral flow in unsaturated homogeneous isotropic sloping soil due to rainfall. Water Resour. Res. 42, W12402 (2006)

    Article  Google Scholar 

  18. Wang, S.: A novel fitted finite volume method for the Black-Scholes equation governing option pricing. IMA J. Numer. Anal. 24(4), 699–720 (2004)

    Article  MathSciNet  Google Scholar 

  19. Wang, S., Shang, S., Fang, Z.: A superconvergence fitted finite volume method for Black-Sholes equation governing European and American options. Numer. Meth. Part. Differ. Equat. 31(4), 1190–1208 (2014)

    Article  Google Scholar 

  20. Zadeh, K.S.: A mass-conservative switching algorithm for modeling fluid flow in variably saturated porous media. J. Comput. Phys. 230, 664–679 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by the Bulgarian National Science Fund under the Project DN 12/4 “Advanced analytical and numerical methods for nonlinear differential equations with applications in finance and environmental pollution”, 2017 and Bilateral Project KP/Russia 06/12 “Numerical methods and algorithms in the theory and applications of classical hydrodynamics and multiphase fluids in porous media”, 2020.

Author information

Authors and Affiliations

  1. University of Ruse, 8 Studentska Street, 7017, Ruse, Bulgaria

    Miglena N. Koleva & Lubin G. Vulkov

Authors
  1. Miglena N. Koleva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lubin G. Vulkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miglena N. Koleva .

Editor information

Editors and Affiliations

  1. IICT-BAS, Sofia, Bulgaria

    Ivan Lirkov

  2. IICT-BAS, Sofia, Bulgaria

    Svetozar Margenov

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Koleva, M.N., Vulkov, L.G. (2022). Fitted Finite Volume Method for Unsaturated Flow Parabolic Problems with Space Degeneration. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2021. Lecture Notes in Computer Science, vol 13127. Springer, Cham. https://doi.org/10.1007/978-3-030-97549-4_60

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-97549-4_60

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-97548-7

  • Online ISBN: 978-3-030-97549-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation