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.
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References
Akrivis G D. Finite difference discretization of the cubic Schrodinger equation. IMA J Numer Anal, 1993, 13: 115–124
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
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
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
Elder K R, Grant M. Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystal. Phys Rev E, 2003, 68: 066703
Elder K R, Katakowski M, Haataja M, et al. Modeling elasticity in crystal growth. Phys Rev Lett, 2002, 88: 245701
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
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
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
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
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
Sun Z Z. A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation. Math Comp, 1995, 64: 1463–1471
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
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
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
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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
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DOI: https://doi.org/10.1007/s11425-015-5025-1
Keywords
- phase field crystal model
- nonlinear evolutionary equation
- finite difference scheme
- solvability
- convergence