Numerical Study of the Cahn-Hilliard Equation in One, Two and Three Dimensions

Abstract

The Cahn-Hilliard equation is related with a number of interesting physical phenomena like the spinodal decomposition, phase separation and phase ordering dynamics. On the other hand this equation is very stiff an the difficulty to solve it numerically increases with the dimensionality and therefore, there are several published numerical studies in one dimension (1D), dealing with different approaches, and much fewer in two dimensions (2D). In three dimensions (3D) there are very few publications, usually concentrate in some specific result without the details of the used numerical scheme. We present here a stable and fast conservative finite difference scheme to solve the Cahn-Hilliard with two improvements: a splitting potential into a implicit and explicit in time part and a the use of free boundary conditions. We show that gradient stability is achieved in one, two and three dimensions with large time marching steps than normal methods.

arXiv comment20 pages with 12 figs. Accepted in the Physica A

References in corpus (9)

• Colossal Magnetoresistant Materials: The Key Role of Phase Separation
• Theory of Phase Ordering Kinetics
• Discovery of microscopic electronic inhomogeneity in the high-Tc superconductor Bi2Sr2CaCu2O8+x
• Inherent Inhomogeneities in Tunneling Spectra of BSCCO Crystals in the Superconducting State
• Nanoscale Phase Separation in Colossal Magnetoresistance Materials: A Lesson for the Cuprates?
• The attainable superconducting Tc in a model of phase coherence by percolation
• Pressure Studies on a High-$T_c$ Superconductor Pseudogap and Critical Temperatures
• A Theory for High-$T_c$ Superconductors Considering Inhomogeneous Charge Distribution
• Upper critical field $H_{c2}$ calculations for the high critical temperature superconductors considering inhomogeneities

Cited by in corpus (22)

• On diffuse interface modeling and simulation of surfactants in two-phase fluid flow
• gPAV-Based Unconditionally Energy-Stable Schemes for the Cahn-Hilliard Equation: Stability and Error Analysis
• Theory of the Fermi Arcs, the Pseudogap, $T_c$ and the Anisotropy in k-space of Cuprate Superconductors
• Phase Separation and the Phase Diagram in Cuprates Superconductors
• Nernst effect and Critical Field in Cuprate Superconductors
• The Disordered Induced Interaction and the Phase Diagram of Cuprates
• Electronic Phase Separation Transition as the Origin of the Superconductivity and the Pseudogap Phase of Cuprates
• Charge segregation model for superconducting correlations in cuprates above $T_c$
• Approximation of Smectic-A liquid crystals
• An Integral Equation Method for the Cahn-Hilliard Equation in the Wetting Problem
• Resistivity Calculations for Cuprate Superconductor Systems using an Electronic Phase Separation
• A framework for the description of charge order, pseudo and superconducting gap, critical temperature and pairing interaction in cuprate superconductors
• Evidence that cuprate superconductors form an array of nanoscopic Josephson junctions
• Efficient and stable time integration of Cahn-Hilliard equations: explicit, implicit and explicit iterative schemes
• Stability of frozen waves in the Modified Cahn--Hilliard model
• A Family of Second-Order Energy-Stable Schemes for Cahn-Hilliard Type Equations
• Measurements and analysis of the upper critical field $H_{c2}$ on an underdoped and overdoped $L{a}_{2-x}S{r}_{x}Cu{O}_4$ compounds
• Charge disorder and variations of $T_c$ in Zn-doped cuprate superconductors
• Unified model of the Hall effect from insulator to overdoped compounds in cuprate superconductors
• The role of the pseudogap in cuprate superconductors revealed by the Hall effect
• Generalization of interlayer tunneling models to cuprate superconductors with charge density waves
• Why the superfluid density tracks $T_{\mathrm{c}}$ in cuprate superconductors?