An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation
TLDR
The unique solvability, energy stability are established for the proposed numerical scheme, and an optimal rate convergence analysis is derived in the $\ell^\infty (0,T; T; H_h^2)$ norm.About:
This article is published in Journal of Computational and Applied Mathematics.The article was published on 2019-12-15 and is currently open access. It has received 127 citations till now. The article focuses on the topics: Finite difference & Stencil.read more
Citations
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Journal ArticleDOI
Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential
TL;DR: The unique solvability and the positivity-preserving property for the second order scheme are proved using similar ideas, in which the singular nature of the logarithmic term plays an essential role.
Book ChapterDOI
The phase field method for geometric moving interfaces and their numerical approximations
Qiang Du,Xiaobing Feng +1 more
TL;DR: In this article, the authors present a holistic overview about the main ideas of phase field modelling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state of the art of numerical approximations of various phase field models with an emphasis on discussing the main idea of numerical analysis techniques.
Journal ArticleDOI
A Third Order Exponential Time Differencing Numerical Scheme for No-Slope-Selection Epitaxial Thin Film Model with Energy Stability
TL;DR: The power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.
Posted Content
A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability
TL;DR: In this paper, a third order accurate exponential time differencing (ETD) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space, was proposed and analyzed.
Journal ArticleDOI
A positivity-preserving, energy stable and convergent numerical scheme for the Cahn–Hilliard equation with a Flory–Huggins–Degennes energy
TL;DR: In this paper, a finite difference algorithm based on a convex splitting technique of the energy functional was proposed for the MMC-TDGL equation, a Cahn-Hilliard equation with a Flory-Huggins-deGennes energy.
References
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Journal ArticleDOI
Free Energy of a Nonuniform System. I. Interfacial Free Energy
John W. Cahn,John E. Hilliard +1 more
TL;DR: In this article, it was shown that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature Tc, and that at a temperature T just below Tc the interfacial free energy σ is proportional to (T c −T) 3 2.
Book
Advanced mathematical methods for scientists and engineers
Carl M. Bender,Steven A. Orszag +1 more
TL;DR: A self-contained presentation of the methods of asymptotics and perturbation theory, methods useful for obtaining approximate analytical solutions to differential and difference equations is given in this paper.
MonographDOI
Numerical analysis of spectral methods : theory and applications
David Gottlieb,Steven A. Orszag +1 more
TL;DR: Spectral Methods Survey of Approximation Theory Review of Convergence Theory Algebraic Stability Spectral Methods Using Fourier Series Applications of algebraic stability analysis Constant Coefficient Hyperbolic Equations Time Differencing Efficient Implementation of Spectral Method as discussed by the authors.