Numerical approximations of Allen-Cahn and Cahn-Hilliard equations

By Jie Shen, Xiaofeng Yang. 2010

1,036 citations recorded by the source corpus.

Stability analyses and error estimates are carried out for a number of commonly used numerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown that all the schemes we considered are either unconditionally energy stable, or conditionally energy stable with reasonable stability conditions in the semi-discretized versions. Error estimates for selected schemes with a spectral-Galerkin approximation are also derived. The stability analyses and error estimates are based on a weak formulation thus the results can be easily extended to other spatial discretizations, such as Galerkin finite element methods, which are based on a weak formulation.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/e72cdd3d7e76dd21c6691816efe3b4e30ebb5ac4

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In Computational Mechanics

1. Xiaofeng Yang
2. Jia Zhao
3. Jie Shen
4. Qi Wang
5. Jiang Yang
6. Tao Tang
7. Steven M. Wise
8. Xiaoming Wang

In Computational Theory and Mathematics

1. Jia Zhao
2. Xiaofeng Yang
3. Qi Wang
4. Jiang Yang
5. Jie Shen
6. Tao Tang
7. Xiaoming Wang
8. Steven M. Wise

In Materials Chemistry

1. Xiaofeng Yang
2. Jia Zhao
3. Qi Wang
4. Jie Shen
5. Jiang Yang
6. Tao Tang
7. Xiaoming Wang
8. Steven M. Wise

Citations in the corpus

• The phase field method for geometric moving interfaces and their numerical approximations (2019, neutral): “Optimal error estimates can be found in [385] for the above approximation and a number of other methods involving spectral approximations in space and semiimplicit or fully-implicit in time schemes.”
• Implicit-Explicit Scheme for the Allen-Cahn Equation Preserves the Maximum Principle (2016, methodology): “For example, we can follow [12] to give a modified scheme: U − U τ + ((U) − U) + β(U − U) = ε(2)DhU, (3.”
• Decoupled, Energy Stable Schemes for Phase-Field Models of Two-Phase Incompressible Flows (2015, methodology): “[18, 7, 33]) to consider the Allen–Cahn and Cahn–Hilliard equations with a truncated double-well potential F̃ (φ).”
• The Cahn-Hilliard Equation with Logarithmic Potentials (2011, neutral): “[126], [134], [135], [136], [138], [139], [141], [142], [143], [144], [145], [146], [147], [152], [158], [191], [200], [201], [203], [208], [209], [211] and [217] for the numerical analysis and simulations of the Cahn-Hilliard equation (and several of its generalizations).”
• Optimal Control for the Convective Cahn–Hilliard Equation in 2D Case (2014, methodology): “For the study of numerical solutions of Cahn–Hilliard equation, we refer the reader to [27,28].”
• Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model (2017, background): “the convex splitting approach (implicit) [41] or the stabilization approach (explicit) [31,39,38].”
• Stabilized Crank-Nicolson/Adams-Bashforth Schemes for Phase Field Models (2013, methodology): “Email addresses: fxlmath gmail. om (X. Feng), ttang math.hkbu.edu.hk (T. Tang), jiangy hkbu.edu.hk (J. Yang) http://www.global-sci.org/eajam 59 c©2013 Global-Science Press and the Cahn-Hilliard equation ∂ u ∂ t =∆ −∆u+ 1 ε2 f (u) , (x , t) ∈ Ω× (0, T] , ∂ u ∂ n |∂Ω = 0 , ∂ ∆u− 1 ε f (u) ∂ n = 0 ,”
• On the maximum principle preserving schemes for the generalized Allen–Cahn equation (2016, neutral): “[15]) un+1−un τ +v ·∇un+1+S(un+1−un)=M(un) ( Δun+1− 1 F ′(un) ) , (2.”
• Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends (2016, methodology): “We will implement the rigorous error analysis in the future work that will follow the same lines as the literatures [10,11,14,31].”
• Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to Thin Film Epitaxy (2012, background): “(We refer to [6, 25, 26, 33, 5, 29] for other related convexity splitting schemes.”
• Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method (2017, methodology): “We will implement the rigorous error analysis in the future work that will follow the same lines as the literatures [7,8,11,30].”