Skip to main content
SpringerLink
Log in

Weighted Time-Semidiscretization Quasilinearization Method for Solving Rihards’ Equation

  • Conference paper
  • First Online:
Large-Scale Scientific Computing (LSSC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11958))

Included in the following conference series:

Abstract

This paper concerns efficient \(\sigma \) - weighted (\(0<\sigma <1\)) time-semidiscretization quasilinearization technique for numerical solution of Richards’ equation. We solve the classical and a new \(\alpha \) - time-fractional (\(0<\alpha <1\)) equation, that models anomalous diffusion in porous media. High-order approximation of the \(\alpha =2(1-\sigma )\) fractional derivative is applied. Numerical comparison results are discussed.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)

    Article  MathSciNet  Google Scholar 

  2. Arbogast, T., Taicher, A.L.: A cell-centered finite difference method for a denerate elliptic equation arising from two-phase mixtures. Comput. Geosci. 21(4), 700–712 (2017)

    Article  Google Scholar 

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

    MATH  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-conservativ numerical solution for the unsaturated flow equation. Water Resour. Res. 26(7), 1483–1496 (1990)

    Article  Google Scholar 

  6. Dimitrov, Y.: Three-point approximation for the Caputo fractional derivative. Commun. Appl. Math. Comput. 31(4), 413–442 (2017)

    MathSciNet  MATH  Google Scholar 

  7. Evans, C., Pollock, S., Rebholz, L.G., Xiao, M.: A proof that Anderson acceleration increases the convergence rate in linearly converging fixed point methods (but not in quadratically converging ones). arXiv:1810.08455v1 [math.NA], 19 October 2018

  8. Gerolymatou, E., Vardoulakis, I., Hilfer, R.: Modelling infiltration by means of a nonlinear fractional diffusion model. J. Phys. D Appl. Phys. 39, 4104–4110 (2006)

    Article  Google Scholar 

  9. Gerolymatou, E., Vardoulakis, I., Hilfer, R.: Simulating the saturation front using a fractional diffusion model. In: 5th GRACM International Congress on Computational Mechanics Limassol, 29 June–1 July 2005 (2005). https://www.icp.uni-stuttgart.de/~hilfer/publikationen/pdfo/ZZ-2005-GRACM-653.pdf

  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. Koleva, M., Vulkov, L.: Two-grid quasilinearization approach to ODEs with applications to model problems in physics and mechanics. Comput. Phys. Commun. 181, 663–670 (2010)

    Article  MathSciNet  Google Scholar 

  12. Koleva, M., Vulkov, L.: Numerical solution of time-fractional Black-Scholes equation. Comput. Appl. Math. 36(4), 1699–1715 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Mitra, K., Pop I.S.: A modified L-scheme to solve nonlinear diffusion problems. Comput. Math. Appl. (in press). https://doi.org/10.1016/j.camwa.2018.09.042

  15. Pachepsky, Y., Timlin, D., Rawls, W.: Generalized Richards’ equation to simulate water transport in saturated soils. J. Hydrol. 272, 3–13 (2003)

    Article  Google Scholar 

  16. Popova, Z., Crevoisier, D., Mailhol, J., Ruelle, P.: Assessment and simulation of water and nitrogen transfer under furrow irrigation: application of hydrus2D model to simulate nitrogen transfer. In: ICID 22nd European Regional Conference 2007, Pavia, Italy, 2–7 September 2007 (2007). https://doi.org/10.13140/2.1.1268.2242

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

    Article  Google Scholar 

  18. 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 research is supported by the Bulgarian National Science Fund under Bilateral Project DNTS/Russia 02/12 from 2018.

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 Lubin G. Vulkov .

Editor information

Editors and Affiliations

  1. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

  2. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Svetozar Margenov

Rights and permissions

Reprints and permissions

Copyright information

© 2020 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. (2020). Weighted Time-Semidiscretization Quasilinearization Method for Solving Rihards’ Equation. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2019. Lecture Notes in Computer Science(), vol 11958. Springer, Cham. https://doi.org/10.1007/978-3-030-41032-2_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-41032-2_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-41031-5

  • Online ISBN: 978-3-030-41032-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation