Skip to main content
SpringerLink
Log in

Operator splitting scheme based on barycentric Lagrange interpolation collocation method for the Allen-Cahn equation

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, we consider the operator splitting scheme based on barycentric Lagrange interpolation collocation method for the two-dimensional Allen-Cahn equation. The original problem is split into linear and nonlinear subproblems: the linear part is solved by barycentric Lagrange interpolation collocation method in space and Crank-Nicolson scheme in time; the nonlinear part is solved analytically due to the availability of a closed-form solution. The error estimates of the proposed scheme are studied. Numerical experiments are carried out to demonstrate the accuracy and efficiency of the two operator splitting schemes.

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

Similar content being viewed by others

References

  1. Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)

    Article  Google Scholar 

  2. Fan, D., Chen, L.Q.: Computer simulation of grain growth using a continuum field model. Acta Mater. 45, 611–622 (1997)

    Article  Google Scholar 

  3. Kobayashi, R.: Modeling and numerical simulations of dendritic crystal growty. Phys. D. 63, 410–423 (1993)

    Article  Google Scholar 

  4. Benes, M., Chalupecky, V., Mikula, K.: Geometrical image segmentation by the Allen-Cahn equation. Appl. Numer. Math. 51, 187–205 (2004)

    Article  MathSciNet  Google Scholar 

  5. Kay, D.A., Tomasi, A.: Color image segmentation by the vector valued Allen-Cahn phase-field model: a multigrid solution. IEEE Trans. Image Process. 18, 2330–2339 (2009)

    Article  MathSciNet  Google Scholar 

  6. Elliott, C.M., Stinner, B.: Computation of two-phase biomembranes with phase dependent material parameters using surface finite element. Commun. Comput. Phys. 13, 325–360 (2013)

    Article  MathSciNet  Google Scholar 

  7. Li, Y., Lee, H.G., Kim, J.: A fast, robust, and accurate operator splitting method for Phase-field simulation of crystal growth. J. Cryst. Growth. 321, 176–182 (2011)

    Article  Google Scholar 

  8. Feng, X.B., Prohl, A.: Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flaws. Numer. Math. 94, 33–65 (2003)

    Article  MathSciNet  Google Scholar 

  9. Golubovic, L., Levandovsky, A., Moldovan, D.: Interface dynamics and far-from-equilibrium phase transitions in multilayer epitaxial growth and erosion on crystal surfaces: Continuum theory insights. East Asian J. Appl. Math. 1, 297–371 (2011)

    Article  MathSciNet  Google Scholar 

  10. Shen, J., Yang, X.F.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Dis. Contin. Dyn. Syst. 28, 1669–1691 (2010)

    Article  MathSciNet  Google Scholar 

  11. Zhai, S.Y., Feng, X.L., He, Y.N.: Numerical simulation of the three dimensional Allen-Cahn equation by the high-order compact ADI method. Comput. Phys. Commun. 185, 2449–2455 (2014)

    Article  Google Scholar 

  12. Feng, X.B., Li, Y.K.: Analysis of symmetric interior penalty discontinuous Galerkin methods for the Allen-Cahn equation and the mean curvature flow. IMA J. Numer. Anal. 35, 1622–1651 (2014)

    Article  MathSciNet  Google Scholar 

  13. Jeong, D., Lee, S., Lee, D., et al.: Comparison study of numerical methods for solving the Allen-Cahn equation. Comput. Mater. Sci. 111, 131–136 (2016)

    Article  Google Scholar 

  14. Li, H.R., Song, Z.Y., Hu, J.Z.: Numerical analysis of a second-order IPDGFE method for the Allen-Cahn equation and the curvature-driven geometric flow. Comput. Math. Appl. 86, 49–62 (2021)

    Article  MathSciNet  Google Scholar 

  15. Li, H.R., Song, Z.Y.: A reduced-order finite element method based on proper orthogonal decomposition for the Allen-Cahn model. J. Math. Anal. Appl. 500, 125103 (2021)

    Article  MathSciNet  Google Scholar 

  16. Li, H.R., Song, Z.Y., Zhang, F.C.: A reduced-order modified finite difference method preserving unconditional energy-stability for the Allen-Cahn equation. Numer. Meth. Part. D. E. 37, 1869–1885 (2021)

    Article  MathSciNet  Google Scholar 

  17. Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968)

    Article  MathSciNet  Google Scholar 

  18. Li, X., Qiao, Z.H., Zhang, H.: Convergence of a fast explicit operator splitting method for the epitaxial growth model with slope selection. SIAM. J. Numer. Anal. 55, 265–285 (2017)

    Article  MathSciNet  Google Scholar 

  19. Zhai, S.Y., Wu, L.Y., Wang, J.Y., Weng, Z.F.: Numerical approximation of the fractional Cahn-Hilliard equation by operator splitting method. Numer. Algorithms. 84, 1155–1178 (2020)

    Article  MathSciNet  Google Scholar 

  20. Weng, Z.F., Deng, Y.F., Zhuang, Q.Q., Zhai, S.Y.: A fast and efficient numerical algorithm for swift-hohenberg equation with a nonlocal nonlinearity. Appl. Math. Lett. 118, 107170 (2021)

    Article  MathSciNet  Google Scholar 

  21. Jeong, D., Kim, J.: An explicit hybrid finite difference scheme for the Allen-Cahn equation. J. Comput. Appl. Math. 340, 247–255 (2018)

    Article  MathSciNet  Google Scholar 

  22. Huang, Y.Q., Yang, W., Wang, H., Cui, J.T.: Adaptive operator splitting finite element method for Allen-Cahn equation. Numer. Meth. Part. D. E. 35, 1290–1300 (2019)

    Article  MathSciNet  Google Scholar 

  23. Weng, Z.F., Tang, L.K.: Analysis of the operator splitting scheme for the Allen-Cahn equation. Numer. Heat. TR. B-Fund. 70, 472–483 (2016)

    Article  Google Scholar 

  24. Liu, H.Y., Huang, J., Pan, Y.B., Zhang, J.: Barycentric interpolation collocation methods for solving linear and nonlinear high-dimensional fredholm integral equations. J. Comput. Appl. Math. 327, 141–154 (2018)

    Article  MathSciNet  Google Scholar 

  25. Liu, F.F., Wang, Y.L., Li, S.G.: Barycentric interpolation collocation method for solving the coupled viscous Burgers equations. Int. J. Comput. Math. 95, 2162–2173 (2018)

    Article  MathSciNet  Google Scholar 

  26. Yi, S.C., Yao, L.Q.: A steady barycentric Lagrange interpolation method for the 2D higher-order time-fractional telegraph equation with nonlocal boundary condition with error analysis. Numer. Meth. Part. D. E. 35, 1694–1716 (2019)

    Article  MathSciNet  Google Scholar 

  27. Liu, H.Y., Huang, J., Zhang, W.: Numerical algorithm based on extended barycentric Lagrange interpolant for two dimensional integro-differential equations. Appl. Math. Comput. 396, 125931 (2021)

    MathSciNet  MATH  Google Scholar 

  28. Deng, Y.F., Weng, Z.F.: Barycentric interpolation collocation method based on Crank-Nicolson scheme for the Allen-Cahn equation. AIMS. Math. 6, 3857–3873 (2021)

    Article  MathSciNet  Google Scholar 

  29. Berrut, J., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46, 501–517 (2004)

    Article  MathSciNet  Google Scholar 

  30. Li, Y.B., Lee, H.G., Jeong, D., Kim, J.: An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation. Comput. Math. Appl. 60, 1591–1606 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is in part supported by the NSF of China (Nos. 11701196, 11701197), the Fundamental Research Funds for the Central Universities (No. ZQN-702), the Natural Science Foundation of Fujian Province (Nos. 2020J01074, 2021J01306) and the Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education (Xiangtan university) (No. 2020ICIP03)

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian, 362021, China

    Yangfang Deng & Zhifeng Weng

Authors
  1. Yangfang Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhifeng Weng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhifeng Weng.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deng, Y., Weng, Z. Operator splitting scheme based on barycentric Lagrange interpolation collocation method for the Allen-Cahn equation. J. Appl. Math. Comput. 68, 3347–3365 (2022). https://doi.org/10.1007/s12190-021-01666-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-021-01666-y

Keywords

Mathematics Subject Classification

Search

Navigation