Advanced operator splitting-based semi-implicit spectral method to solve the binary phase-field crystal equations with variable coefficients

Journal of Computational Physics

Volumn 228, Issue 5, 2009, Pages 1612-1623

  Tegze, György ^a   Bansel, Gurvinder ^a   Tóth, Gyula I ^b   Pusztai, Tamás ^b   Fan, Zhongyun ^a   Gránásy, László ^a  

^a BRUNEL UNIVERSITY   (United Kingdom)

^b Research Institute of Solid State Physics and Optics   (Hungary)

Author keywords

Operator splitting; Phase field crystal; Spectral methods

Indexed keywords

FINITE DIFFERENCE METHOD; INTEGRATION TESTING;

BINARY PHASIS; DISSIPATIVE DYNAMICS; IMPLICIT SOLUTIONS; OPERATOR SPLITTING METHOD; OPERATOR-SPLITTING; PHASE-FIELD CRYSTAL MODELS; PHASE-FIELD CRYSTALS; SEMI-IMPLICIT; SPECTRAL METHODS; VARIABLE COEFFICIENTS;

PROGRAM PROCESSORS;

EID: 58249131435     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2008.11.011     Document Type: Article

References (45)

1. Gunton J.D., San Miguel M., and Sahni P.S. In: Domb C., and Lebowitz J.L. (Eds). Phase Transitions and Critical Phenomena vol. 8 (1983), Academic Press, New York 267
2. Langer J.S. In: Godrèche C. (Ed). Solids Far from Equilibrium (1992), Cambridge University Press, Cambridge 297
3. Bray A.J. Theory of phase-ordering kinetics. Adv. Phys. 43 (1994) 357
4. Cahn J.W., and Hilliard J.E. Free energy of a nonuniform system. I. Interface free energy. J. Chem. Phys. 28 (1958) 258
5. Allen S.M., and Cahn J.W. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1085
6. Shih W.H., Wang Z.Q., Zeng X.C., and Stroud D. Ginzburg-Landau theory for the solid-liquid interface of bcc elements. Phys. Rev. A 35 (1987) 2611
7. Rogers T.M., Elder K.R., and Desai R.C. Numerical study of the late stages of spinodal decomposition. Phys. Rev. B 37 (1988) 9638
8. Oono Y., and Puri S. Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling. Phys. Rev. A 38 (1998) 434
9. Wu K.-A., Karma A., Hoyt J.J., and Asta M. Ginzburg-Landau theory of crystalline anisotropy for bcc-liquid interfaces. Phys. Rev. B 73 (2006) 094101
10. Elder K.R., Katakowski M., Haataja M., and Grant M. Modeling elasticity in crystal growth. Phys. Rev. Lett. 88 (2002) 245701
11. Elder K.R., and Grant M. Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys. Rev. E 70 (2004) 051605
12. For review on classical DFT of crystallization see:. Oxtoby D.W. Crystallization of liquids: a density functional approach. In: Hansen J.P., Levesque D., and Zinn-Justin J. (Eds). Liquids, Freezing, and Glass Transition vol. I (1991), North Holland, Amsterdam 145
13. Swift J., and Hohenberg P.C. Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15 (1977) 319
14. Brazovsky S.A. Phase transition of an isotropic system to an inhomogeneous state. Zh. Eksp. Teor. Fiz. 68 (1975) 175
15. Provatas N., Dantzig J.A., Athreya B., Chan P., Stefanovic P., Goldenfeld N., and Elder K.R. Using the phase-field crystal method in the multi-scale modeling of microstructure evolution. JOM 59 (2007) 83
16. Elder K.R., Provatas N., Berry J., Stefanovic P., and Grant M. Phase-field crystal modeling and classical density functional theory of freezing. Phys. Rev. B. 75 (2007) 064107
17. Cheng M., and Warren J.A. An efficient algorithm for solving the phase field crystal method. J. Comput. Phys. 227 (2008) 6241
18. Goldenfeld N., Athreya B.P., and Dantzig J.A. Renormalization group approach to multiscale simulation of polycrystalline materials using the phase field crystal model. Phys. Rev. E 72 (2005) 020601
19. Provatas N., Greenwood M., Athreya B., and Goldenfeld N. Multiscale Modeling of Solidification: Phase-Field Methods to Adaptive Mesh Refinement. Int. J. Mod. Phys. B 59 (2005) 4525
20. Athreya B.P., Goldenfeld N., Dantzig J.A., Greenwood M., and Provatas N. Adaptive mesh computation of polycrystalline pattern formation using a renormalization-group reduction of the phase-field crystal model. Phys. Rev. E 76 (2007) 056706
21. Strang G. On the construction and comparison of different splitting schemes. SIAM J. Numer. Anal. 5 (1968) 506
22. Marchuk G.I. Methods of Splitting (1988), Nauka, Moscow
23. Christov C.I., and Marinova R.S. Implicit vectorial operator splitting for incompressible Navier - Stokes equations in primitive variables. J. Comput. Technol. 6 (2001) 92
24. Mimura M., Nakaki T., and Tomoeda K. A numerical approach to interface curves for some nonlinear diffusion equations. Jpn. J. Appl. Math. 1 (1984) 93
25. Jakobsen E.R., Karlsen K.H., and Risebro N.H. On the convergence rate of operator splitting for Hamilton-Jacobi equations with source terms. SIAM J. Numer. Anal. 39 (2001) 499
26. Karlsen K.H., and Risebro N.H. Unconditionally stable methods for Hamilton-Jacobi equations. J. Comput. Phys. 180 (2002) 710
27. Karlsen K.H., Lie K.-A., Natvig J.R., Nordhaug H.F., and Dahle H.K. Operator splitting methods for systems of convection-diffusion equations: nonlinear error mechanisms and correction strategies. J. Comput. Phys. 173 (2001) 636
28. Marinova R.S., Christov C.I., and Marinov T.T. A fully coupled solver for incompressible Navier-Stokes equations using operator splitting. Int. J. Comput. Fluid Dynam. 17 (2003) 371
29. Faragó I., and Havasi A. On the convergence and local splitting error of different splitting schemes. Prog. Comput. Fluid Dynam. 5 (2005) 495
30. Csomós P., and Faragó I. Error analysis of the numerical solution of split differential equations. Math. Comput. Model. 48 (2008) 1090
31. Woodhead-Galloway J., and Gaskel T. Structure of mixtures of simple fluids. J. Phys. C 1 (1968) 1472
32. Musicki D. On canonical formalism in field theory with derivatives of higher order-canonical transformations. Physica A 11 (1978) 39
33. Hundsdorfer W., and Verwer J.G. Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations (2003), Springer-Verlag, Berlin
34. Faragó I. Splitting methods and their application to the abstract Cauchy problems. Lecture Notes in Computer Science vol. 3401 (2005), Springer-Verlag, Berlin 35
35. P. Csomós, I. Faragó, Á. Havasi, Weighted sequential splittings and their analysis, in: Proceedings of NMCM-2002, Comput. Math. Appl., vol. 50, 2005, p. 1017.
36. Levin J.G., Iskandarani M., and Haidvogel D.B. A spectral filtering procedure for eddy-resolving simulations with a spectral element ocean model. J. Comput. Phys. 137 (1997) 130
37. Garcia-Ojalvo J., Sancho J.M., and Remirez-Piscina L. Generation of spatiotemporal colored noise. Phys. Rev. A 46 (1992) 4670
38. Sancho J.M., Garcia-Ojalvo J., and Guo H. Non-equilibrium Ginzburg-Landau model driven by colored noise. Physica D 113 (1998) 331
39. Frigo M., and Johnson S.G. The design and implementation of FFTW3. Proc. IEEE 93 (2005) 216
40. Budd C., Koch O., and Weinmüller E. Computation of self-similar solution profiles for the nonlinear Schrödinger equation. Computing 77 (2006) 335
41. Davidchack R.L., and Laird B.B. Simulation of the hard-sphere crystal-melt interface. J. Chem. Phys. 108 (1998) 9452
42. Ramalingam H., Asta M., van de Walle A., and Hoyt J.J. Atomic-scale study of equilibrium solute adsorption at alloy solid-liquid interfaces. Interf. Sci. 10 (2002) 149
43. Sun D.Y., Asta M., and Hoyt J.J. Crystal-melt free energies and mobilitites in fcc and bcc Fe. Phys. Rev. B 69 (2004) 174103
44. Morris J.R. Complete mapping of the anisotropy of the crystal-melt interface in Al. Phys. Rev. B 66 (2002) 144104
45. Gottlieb D., and Orszag S.A. Numerical Analysis of Spectral Methods: Theory and Application (1977), SIAM CBMS, Philadelphia, PA