Abstract
The critical dynamics of a spatially inhomogeneous system are analyzed with allowance for local nonequilibrium, which leads to a singular perturbation in the equations due to the appearance of a second time derivative. An extension is derived for the Eyre theorem, which holds for classical critical dynamics described by first-order equations in time and based on the local equilibrium hypothesis. It is shown that gradient-stable numerical algorithms can also be constructed for second-order equations in time by applying the decomposition of the free energy into expansive and contractive parts, which was suggested by Eyre for classical equations. These gradient-stable algorithms yield a monotonically nondecreasing free energy in simulations with an arbitrary time step. It is shown that the gradient stability conditions for the modified and classical equations of critical dynamics coincide in the case of a certain time approximation of the inertial dynamics relations introduced for describing local nonequilibrium. Model problems illustrating the extended Eyre theorem for critical dynamics problems are considered.
Similar content being viewed by others
References
B. I. Halperin, P. C. Hohenberg, and S.-K. Ma, “Renormalization-Group Methods for Critical Dynamics: I. Recursion Relations and Effects of Energy Conservation,” Phys. Rev. 10, 139–153 (1974).
A. J. Bray, “Theory of Phase Ordering Kinetics,” Adv. Phys. 43, 357–459 (1994).
S. M. Allen and J. W. Cahn, “A Macroscopic Theory for Antiphase Boundary Motion and Its Application To Antiphase Domain Coarsening,” Acta Metal. 27, 1085–1095 (1979).
J. W. Cahn and J. E. Hilliard, “Free Energy of a Nonuniform System: I. Interfacial Free Energy,” J. Chem. Phys. 28, 258–267 (1958).
J. Swift and P. C. Hohenberg, “Hydrodynamic Fluctuations at Convective Instability,” Phys. Rev. A 15, 319–328 (1977).
K. R. Elder and M. Grant, “Modeling Elastic and Plastic Deformations in Nonequilibrium Processing Using Phase Field Crystals,” Phys. Rev. E 70, 051605 (2004).
J. D. Gunton, R. Toral, and A. Chakrabarti, “Numerical Studies of Phase Separation,” Phys. Scr. 33, 12–19 (1990).
N. Vladimirova, A. Malagoli, and R. Mauri, “Diffusion-Driven Phase Separation of Deeply Quenched Mixtures,” Phys. Rev. E 58, 7691 (1998).
S. Wise, J. Kim, and J. Lowengrub, “Solving the Regularized, Strongly Anisotropic Cahn-Hilliard Equation by An Adaptive Nonlinear Multigrid Method,” Comput. Phys. 226, 414–446 (2007).
I. Singer-Loginova and H. M. Singer, “The Phase Field Technique for Modeling Multiphase Materials,” Rep. Progress. Phys. 71, 106501 (2008).
H. Emmerich, “Advances of and by Phase-Field Modeling in Condensed-Matter Physics,” Adv. Phys. 57(1), 1–87 (2008).
T. M. Rogers, K. R. Elder, and R. C. Desai, “Numerical Study of the Late Stages of Spinodal Decomposition,” Phys. Rev. B 37, 9638 (1988).
D. J. Eyre, Preprint (http://www.math.utah.edu/~eyre/research/methods/stable.ps).
D. J. Eyre, “Unconditionally Gradient-stable Time Step Marching the Cahn-Hilliard Equation,” Computational and Mathematical Models of Microstructural Evolution (Materials Res. Soc. Warrendale, PA, 1998), pp. 39–46.
B. P. Vollmayer-Lee and A. D. Rutenberg, “Fast and Accurate Coarsening Simulation with an Unconditionally Stable Time Step,” Phys. Rev. E 68, 66703 (2003).
M. Cheng and J. A. Warren, “Controlling the Accuracy of Unconditionally Stable Algorithms in the Cahn-Hilliard Equation,” Phys. Rev. E 75, 017702 (2007).
M. Cheng and J. A. Warren, “An Efficient Algorithm for Solving the Phase Field Crystal Model,” J. Comput. Phys. 227, 6241–6248 (2008).
D. Jou, J. Casas-Vazquez, and G. Lebon, Extended Irreversible Thermodynamics, 4th ed. (Springer-Verlag, Berlin, 2010; Moscow, RKhD, 2006).
D. Herlach, R. Galenko, and D. Holland-Moritz, Metastable Solids from Undercooled Melts (Elsevier, Amsterdam 2007; Moscow, RKhD, 2010).
P. Galenko, “Phase-Field Model with Relaxation of the Diffusion Flux in Nonequilibrium Solidification of a Binary System,” Phys. Lett. A 287(3, 4), 190–197 (2001).
P. Galenko and D. Jou, “Diffuse-Interface Model for Rapid Phase Transformations in Nonequilibrium Systems,” Phys. Rev. E 71, 046125 (2005).
P. Galenko and V. Lebedev, “Analysis of the Dispersion Relation in Spinodal Decomposition of a Binary System,” Philos. Mag. Lett. 87, 821–827 (2007).
P. Galenko, D. Danilov, and V. Lebedev, “Phase-Field-Crystal and Swift-Hohenberg Equations with Fast Dynamics,” Phys. Rev. E 79, 051110 (2009).
N. Lecoq, H. Zapolsky, and P. Galenko, “Evolution of the Structure Factor in a Hyperbolic Model of Spinodal Decomposition,” Eur. Phys. J. Special Topics 117, 165–175 (2009).
P. Galenko, “Solute Trapping and Diffusionless Solidification in a Binary System,” Phys. Rev. E 76, 031606 (2007).
V. G. Lebedev, E. V. Abramova, D. A. Danilov, and P. K. Galenko, “Phase-Field Modeling of Solute Trapping: Comparative Analysis of Parabolic and Hyperbolic Models,” Int. J. Math. Res. 101, 473–479 (2010).
M. Asta, H. Harith, Y. Yang, S. Deyan, et al., “Molecular Dynamics Simulations of Solute Trapping and Solute Drag,” Abstracts of Conference PTM-2010 (Avignon, France, 2010), p. 29.
L. D. Landau, “Scattering of X-Rays by Crystals near the Curie Point,” Zh. Eksp. Teor. Fiz. 7 1232 (1937).
V. L. Ginzburg and L. D. Landau, “On the Theory of Superconductivity,” Zh. Eksp. Teor. Fiz. 20, 1064 (1950).
S. A. Brazovskii, “Phase Transition of an Isotropic System to an Inhomogeneous State,” Zh. Eksp. Teor. Fiz. 68, 175–185 (1975).
P. Galenko and D. Jou, “Kinetic Contribution to the Fast Spinodal Decomposition Controlled by Diffusion,” Physica 388, 3113–3123 (2009).
P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953; Inostrannaya Literatura, Moscow, 1960), Vol. 1.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © P.K. Galenko, V.G. Lebedev, and A.A. Sysoeva, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 6, pp. 1148–1165.
Rights and permissions
About this article
Cite this article
Galenko, P.K., Lebedev, V.G. & Sysoeva, A.A. Gradient stability of numerical algorithms in local nonequilibrium problems of critical dynamics. Comput. Math. and Math. Phys. 51, 1074–1090 (2011). https://doi.org/10.1134/S0965542511060078
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542511060078