Skip to main content
SpringerLink
Log in

Two finite difference schemes for the phase field crystal equation

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

The phase field crystal (PFC) model is a nonlinear evolutionary equation that is of sixth order in space. In the first part of this work, we derive a three level linearized difference scheme, which is then proved to be energy stable, unique solvable and second order convergent in L 2 norm by the energy method combining with the inductive method. In the second part of the work, we analyze the unique solvability and convergence of a two level nonlinear difference scheme, which was developed by Zhang et al. in 2013. Some numerical results with comparisons are provided.

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

Similar content being viewed by others

References

  1. Akrivis G D. Finite difference discretization of the cubic Schrodinger equation. IMA J Numer Anal, 1993, 13: 115–124

    Article  MathSciNet  MATH  Google Scholar 

  2. Akrivis G D, Dogalis V A, Karakashian O A. On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrodinger equation. Numer Math, 1991, 59: 31–53

    Article  MathSciNet  MATH  Google Scholar 

  3. Baskaran A, Hu Z, Lowengrub J S, et al. Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation. J Comput Phys, 2013, 250: 270–292

    Article  MathSciNet  Google Scholar 

  4. Baskaran A, Lowengrub J S, Wang C, et al. Convergence of a second order convex splitting scheme for the modified phase field crystal equation. SIAM J Numer Anal, 2013, 51: 2851–2873

    Article  MathSciNet  MATH  Google Scholar 

  5. Elder K R, Grant M. Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystal. Phys Rev E, 2003, 68: 066703

    Article  Google Scholar 

  6. Elder K R, Katakowski M, Haataja M, et al. Modeling elasticity in crystal growth. Phys Rev Lett, 2002, 88: 245701

    Article  Google Scholar 

  7. Galenko P K, Gomez H, Kropotin N V, et al. Unconditionally stable method and numerical solution of the hyperbolic phase-field crystal equation. Phys Rev E, 2013, 88: 013310

    Article  Google Scholar 

  8. Gomez H, Nogueira X. An unconditionally energy-stable method for the phase field crystal equation. Comput Methods Appl Mech Engrg, 2012, 249–252: 52–61

    Article  MathSciNet  Google Scholar 

  9. Hu Z, Wise S M, Wang C, et al. Stable and efficient finite-difference nonlinear multigrid schemes for the phase field crystal equation. J Comput Phys, 2009, 228: 5323–5339

    Article  MathSciNet  MATH  Google Scholar 

  10. Li J, Sun Z Z, Zhao X. A Three Level Linearized Compact Difference Scheme for the Cahn-Hilliard Equation. Sci China Math, 2012, 55: 805–826

    Article  MathSciNet  MATH  Google Scholar 

  11. Provatas N, Dantzig J, Athreya B, et al. Stefanovic, N. Goldenfeld, K. Elder, Using the phase-field crystal method in the multi-scale modeling of microstructure evolution. J Miner Met Mater Soc, 2007, 59: 83–90

    Article  Google Scholar 

  12. Sun Z Z. A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation. Math Comp, 1995, 64: 1463–1471

    MathSciNet  MATH  Google Scholar 

  13. Wang C, Wise S M. An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. SIAM J Numer Anal, 2011, 49: 945–969

    Article  MathSciNet  MATH  Google Scholar 

  14. Wise S M, Wang C, Lowengrub J S. An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J Numer Anal, 2009, 47: 2269–2288

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhang Z R, Ma Y, Qiao Z H. An adaptive time-stepping strategy for solving the phase field crystal model. J Comput Phys, 2013, 249: 204–215

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, China

    HaiYan Cao & ZhiZhong Sun

Authors
  1. HaiYan Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. ZhiZhong Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to ZhiZhong Sun.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, H., Sun, Z. Two finite difference schemes for the phase field crystal equation. Sci. China Math. 58, 2435–2454 (2015). https://doi.org/10.1007/s11425-015-5025-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-015-5025-1

Keywords

MSC(2010)

Search

Navigation