Skip to main content
SpringerLink
Log in

Numerical simulation of moving boundary problem by modified Keller box method with boundary immobilisation technique

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

A modified Keller box (MKB) method is applied to the moving boundary problem (MBP) based on the boundary immobilisation technique. MBP is the modelling of the melting or solidification process. The mathematical importance of this problem is that the boundary of the domain is also unknown. The moving front position depends on time; hence this problem is inherently non-linear. Simulation time is used to evaluate the computational complexity of the schemes. The proposed scheme is compared to the existing schemes in the literature regarding the accuracy and simulation time. In both space and time, the proposed scheme has a second-order accuracy. For the known boundary, the constant boundary condition is taken and the proposed numerical algorithm is validated with the corresponding similarity solution. The MKB method provides good agreement with the similarity solution and also confirms that the computational rate of convergence of our scheme is two. This paper gives an idea of the MKB scheme for an MBP. This mathematical framework can be extended to 2D and 3D MBPs.

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

Similar content being viewed by others

References

  1. G Lamé and B P Clapeyron, Ann. Chim. Phys. 47, 250 (1831)

    Google Scholar 

  2. J Stefan, Ann. Phys. 278, 269 (1891)

    Article  Google Scholar 

  3. L I Rubinšteĭn, The Stefan problem (American Mathematical Society, Rhode Island, 1971)

    Google Scholar 

  4. F Neumannn, Reimann–Weber 2, 121 (1912)

    Google Scholar 

  5. H Go Landau, Quart. Appl. Math. 8, 81 (1950)

  6. S Kutluay, A R Bahadir and A Özdeş, J. Comput. Appl. Math. 81, 135 (1997)

    Article  MathSciNet  Google Scholar 

  7. P C Meek and J Norbury, SIAM J. Numer. Anal. 21, 883 (1984)

  8. S Savović and J Caldwell, Int. J. Heat Mass Transf. 46, 2911 (2003)

    Article  Google Scholar 

  9. S L Mitchell and M Vynnycky, Appl. Math. Comput. 215, 1609 (2009)

    MathSciNet  Google Scholar 

  10. N Sadoun, E-K Si-Ahmed, P Colinet and J Legrand, C. R. Mécanique 340, 501 (2012)

    Article  ADS  Google Scholar 

  11. S Kutluay, Appl. Math. Comput. 168, 1159 (2005)

    MathSciNet  Google Scholar 

  12. M Vynnycky and S L Mitchell, J. Comput. Appl. Math. 276, 98 (2015)

    Article  MathSciNet  Google Scholar 

  13. H Karabenli, A Esen and E Nesligul Aksan, New Trend Math. Sci. 4, 175 (2016)

    Article  Google Scholar 

  14. A Kumar, A K Singh and Rajeev, J King Saud. Univ. Sci. 32, 384 (2020)

  15. J Caldwell and Y Y Kwan, Commun. Numer. Methods Eng. 20, 535 (2004)

    Article  Google Scholar 

  16. J Crank, Quart. J. Mech. Appl. Math. 10, 220 (1957)

  17. A Friedman, Partial differential equations of parabolic type (Courier Dover Publications, New York, 2008)

  18. M K Kadalbajoo and A Awasthi, Appl. Math. Comput. 183, 42 (2006)

  19. H B Keller, Numerical solution of partial differential equations-II (Elsevier, Amsterdam, 1971)

  20. R S Varga, Iterative analysis (Springer, Berlin, 1962)

  21. C S Rao and Manisha, Appl. Math. Comput. 321, 401 (2018)

  22. N Kumar, S Dash, AK Tyagi and B Raj, Pramana – J. Phys. 71, 529 (2008)

    Article  ADS  Google Scholar 

  23. P Mishra, M R Acharya and S Panda, Pramana – J. Phys. 95, 1 (2021)

    Article  Google Scholar 

  24. M Nazeer, N Ali, F Ahmad and M Latif, Pramana – J. Phys. 94, 1 (2020)

    Article  Google Scholar 

Download references

Acknowledgements

The first author would like to thank the Council of Scientific and Industrial Research (CSIR), India for financial support in the form of a research grant. The authors also would like to thank the DST, Government of India, for providing support to carry out this work under the scheme ‘FIST’ (No. SR/FST/MS-I/2019/40) and Council of Scientific and Industrial Research, India (Grant No. 09/874(0023)/2017-EMR-I). The authors are also very much thankful to the anonymous reviewers for their valuable suggestions and comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Calicut, Kozhikode, 673 601, India

    V P Rabeeb Ali & Ashish Awasthi

Authors
  1. V P Rabeeb Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ashish Awasthi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ashish Awasthi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rabeeb Ali, V.P., Awasthi, A. Numerical simulation of moving boundary problem by modified Keller box method with boundary immobilisation technique. Pramana - J Phys 97, 34 (2023). https://doi.org/10.1007/s12043-022-02506-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12043-022-02506-9

Keywords

PACS Nos

Search

Navigation