Skip to main content
SpringerLink
Log in

A RBF-WENO Finite Difference Scheme for Non-linear Degenerate Parabolic Equations

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

Abstract

In this research, a radial basis functions weighted essentially non-oscillatory (RBF-WENO) scheme in the framework of finite difference is developed for solving non-linear degenerate parabolic (NDP) equations which may contain discontinuous solutions. The traditional WENO schemes for solving NDP equations are based on the polynomial interpolation. In this paper, a non-polynomial WENO finite difference scheme in order to enhance the local accuracy is proposed. For a non-polynomial interpolation basis, the infinitely smooth RBFs such as the multi-quadratic are adopted. By optimizing the free parameter in RBFs, the local error generated by the RBFs interpolation can be easily controlled. The formulation of scheme will be described in detail and numerical tests have been prepared to demonstrate the efficiency of the scheme in one and two-dimension. The numerical results demonstrate that the developed non-polynomial WENO schemes enhance the local accuracy and give sharper solution profile than the original WENO based on the polynomial interpolation for solving NDP equations.

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

Access this article

Log in via an institution

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

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Abedian, R.: WENO schemes for multidimensional nonlinear degenerate parabolic PDEs. Iran. J. Numer. Anal. Optim. 8, 41–62 (2018)

    MATH  Google Scholar 

  2. Abedian, R.: A new high-order weighted essentially non-oscillatory scheme for non-linear degenerate parabolic equations. Numer. Methods Partial Differ. Equ. 37, 1317–1343 (2021)

    Article  MathSciNet  Google Scholar 

  3. Abedian, R.: WENO-Z schemes with Legendre basis for non-linear degenerate parabolic equations. Iran. J. Math. Sci. Inform. 16, 125–143 (2021)

    MathSciNet  MATH  Google Scholar 

  4. Abedian, R., Adibi, H., Dehghan, M.: A high-order weighted essentially non-oscillatory (WENO) finite difference scheme for nonlinear degenerate parabolic equations. Comput. Phys. Commun. 184, 1874–1888 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Abedian, R., Dehghan, M.: A high-order weighted essentially nonoscillatory scheme based on exponential polynomials for nonlinear degenerate parabolic equations. Numer. Methods Partial Differ. Equ. 38, 970–996 (2022)

    Article  MathSciNet  Google Scholar 

  6. Alt, H.W., Luckhaus, S.: Quasilinear elliptic–parabolic differential equations. Math. Z. 183, 311–341 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  7. Arbogast, T., Huang, C.S., Zhao, X.: Finite volume WENO schemes for nonlinear parabolic problems with degenerate diffusion on non-uniform meshes. J. Comput. Phys. 399, 108,921-108,923 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Aregba-Driollet, D., Natalini, R., Tang, S.: Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems. Math. Comput. 73, 63–94 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Balsara, D., Shu, C.W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bessemoulin-Chatard, M., Filbet, F.: A finite volume scheme for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 34, B559–B583 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Buhman, M.D.: Radial Basis Functions-Theory & Implementations. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  12. Capdeville, G.: A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes. J. Comput. Phys. 227, 2977–3014 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Castro, M., Costa, B., Don, W.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230, 1766–1792 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cavalli, F., Naldi, G., Puppo, G., Semplice, M.: High order relaxation schemes for nonlinear degenerate diffusion problems. SIAM J. Numer. Anal. 45, 2098–2119 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Christlieb, A., Guo, W., Jiang, Y., Yang, H.: Kernel based high order explicit unconditionally stable scheme for nonlinear degenerate advection–diffusion equations. J. Sci. Comput. 82, 52 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Duyn, C.J., Peletier, L.A.: Nonstationary filtration in partially saturated porous media. Arch. Ration. Mech. Anal. 78, 173–198 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fan, S.J.: A new extracting formula and a new distinguishing means on the one variable cubic equation. J. Hainan Norm. Univ. Nat. Sci. Ed. 2, 91–98 (1989). (in Chinese)

    Google Scholar 

  18. Guo, J., Jung, J.H.: Radial basis function ENO and WENO finite difference methods based on the optimization of shape parameters. J. Sci. Comput. 70, 551–575 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guo, J., Jung, J.H.: A RBF-WENO finite volume method for hyperbolic conservation laws with the monotone polynomial interpolation method. Appl. Numer. Math. 112, 27–50 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Henrick, A.K., Aslam, T.D., Powers, J.M.: Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207, 542–567 (2005)

    Article  MATH  Google Scholar 

  21. Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jiang, Y.: High order finite difference multi-resolution WENO method for nonlinear degenerate parabolic equations. J. Sci. Comput. 86, 16 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kurganov, A., Tadmor, E.: New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. J. Comput. Phys. 160, 241–282 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Larsson, E., Fornberg, B.: Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. Comput. Math. Appl. 49, 103–130 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lee, Y., Yoon, J., Micchelli, C.: On convergence of flat multivariate interpolation by translation kernels with finite smoothness. Constr. Approx. 40, 37–60 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, Y., Shu, C.W., Zhang, M.: High order finite difference WENO schemes for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 33, 939–965 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lu, Y., Jger, W.: On solutions to nonlinear reaction–diffusion–convection equations with degenerate diffusion. J. Differ. Equ. 170, 1–21 (2001)

    Article  MathSciNet  Google Scholar 

  29. Otto, F.: L1-contraction and uniqueness for quasilinear elliptic–parabolic equations. J. Differ. Equ. 131, 20–38 (1996)

    Article  MATH  Google Scholar 

  30. Shi, J., Hu, C., Shu, C.W.: A technique of treating negative weights in WENO schemes. J. Comput. Phys. 175, 108–127 (2002)

    Article  MATH  Google Scholar 

  31. Shu, C.W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51, 82–126 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  33. Vàzquez, J.L.: The Porous Medium Equation Mathematical Theory. Oxford University Press Inc, New York (2007)

    MATH  Google Scholar 

  34. Zhang, Q., Wu, Z.: Numerical simulation for porous medium equation by local discontinuous Galerkin finite element method. J. Sci. Comput. 38, 127–148 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhu, J., Qiu, J.: A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110–121 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are very thankful to the reviewers for carefully reading the paper, their comments and suggestions have improved the quality of the paper.

Funding

No funding was received for conducting this study.

Author information

Authors and Affiliations

  1. School of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran

    Rooholah Abedian

  2. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran

    Mehdi Dehghan

Authors
  1. Rooholah Abedian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mehdi Dehghan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rooholah Abedian.

Ethics declarations

Competing interests

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abedian, R., Dehghan, M. A RBF-WENO Finite Difference Scheme for Non-linear Degenerate Parabolic Equations. J Sci Comput 93, 60 (2022). https://doi.org/10.1007/s10915-022-02022-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-022-02022-3

Keywords

Mathematics Subject Classification

Search

Navigation