Finite difference approximate solutions for the Cahn‐Hilliard equation

Khiari, Noomen; Achouri, Talha; Mohamed, M.; Omrani, Khaled

doi:10.1002/num.20189

Cited by 33 publications

(18 citation statements)

References 28 publications

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“…However, because of nonlinear nature of the Cahn-Hilliard equation, implicit schemes require nonlinear solvers. Probably the most utilized implicit time schemes in this context are Euler Backward and Crank-Nicholson methods [27,22,23].…”

Section: Temporal Discretizationmentioning

confidence: 99%

Computationally efficient solution to the Cahn–Hilliard equation: Adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem

Wodo

Ganapathysubramanian

2011

Journal of Computational Physics

134

106

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a b s t r a c tWe present an efficient numerical framework for analyzing spinodal decomposition described by the Cahn-Hilliard equation. We focus on the analysis of various implicit time schemes for two and three dimensional problems. We demonstrate that significant computational gains can be obtained by applying embedded, higher order Runge-Kutta methods in a time adaptive setting. This allows accessing time-scales that vary by five orders of magnitude. In addition, we also formulate a set of test problems that isolate each of the sub-processes involved in spinodal decomposition: interface creation and bulky phase coarsening. We analyze the error fluctuations using these test problems on the split form of the Cahn-Hilliard equation solved using the finite element method with basis functions of different orders. Any scheme that ensures at least four elements per interface satisfactorily captures both sub-processes. Our findings show that linear basis functions have superior error-to-cost properties.This strategy -coupled with a domain decomposition based parallel implementationlet us notably augment the efficiency of a numerical Cahn-Hillard solver, and open new venues for its practical applications, especially when three dimensional problems are considered. We use this framework to address the isoperimetric problem of identifying local solutions in the periodic cube in three dimensions. The framework is able to generate all five hypothesized candidates for the local solution of periodic isoperimetric problem in 3D -sphere, cylinder, lamella, doubly periodic surface with genus two (Lawson surface) and triply periodic minimal surface (P Schwarz surface).

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Section: Temporal Discretizationmentioning

confidence: 99%

Computationally efficient solution to the Cahn–Hilliard equation: Adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem

Wodo

Ganapathysubramanian

2011

Journal of Computational Physics

134

106

View full text Add to dashboard Cite

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“…(1.1) have been considered by Choo and Chung [32] and by Choo et al [33] using a nonlinear conservative difference scheme for two-dimensional problem. Another Crank-Nicolson type method has been developed by Khiari et al [34] for (1.1) in one dimension. Mello et al [36] have presented a stable and fast conservative finite difference scheme to solve (1.1) with two improvements.…”

Section: Introductionmentioning

confidence: 99%

“…There have been many algorithms developed and simulations performed for the C-H equations, using Finite Element Methods [18][19][20][21][22][23][24][25], Discontinuous Galerkin Techniques [26][27][28], Finite Difference Schemes [29][30][31][32][33][34][35][36], Spectral Methods [37,38], Collocation Techniques [39][40][41], Adomian Decomposition Procedure [42], m-transform [43] and etc.…”

Section: Introductionmentioning

confidence: 99%

A numerical method based on the boundary integral equation and dual reciprocity methods for one-dimensional Cahn–Hilliard equation

Dehghan

Mirzaei

2009

Engineering Analysis with Boundary Elements

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“…This error estimate is necessary to make the error term associated with the nonlinear convection have a non-positive inner product with the corresponding error test function, which is crucial to the convergence analysis. In particular, we note that, although the ∞ (0, T ; H 1 ) error estimates have been available for the pure Cahn-Hilliard equation in the existing literature [3,17,24,29], an 2 (0, T ; H 3 ) error estimate remains open for the finite element approximation applied to the related PDE systems, in the authors' knowledge.…”

Section: Definition 11 Definementioning

confidence: 99%

Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system

et al. 2016

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We present and analyze a mixed finite element numerical scheme for the Cahn-Hilliard-Hele-Shaw equation, a modified Cahn-Hilliard equation coupled with the Darcy flow law. This numerical scheme was first reported in Feng and Wise (SIAM J Numer Anal 50:1320-1343, 2012), with the weak convergence to a weak solution proven. In this article, we provide an optimal rate error analysis. A convex splitting approach is taken in the temporal discretization, which in turn leads to the unique solvability and unconditional energy stability. Instead of the more standard ∞ (0, T ; L 2 ) ∩ 2 (0, T ; H 2 ) error estimate, we perform a discrete ∞ (0, T ; H 1 ) ∩ 2 (0, T ; H 3 ) error estimate for the phase variable, through an L 2 inner product with the numerical error function associated with the chemical potential. As a result, an unconditional convergence (for the time step τ in terms of the spatial resolution h) is derived. The nonlinear analysis is accomplished with the help of a discrete Gagliardo-Nirenberg type inequality in the finite element space, gotten by introducing a discrete Laplacian h of the numerical solution, such that h φ ∈ S h , for every φ ∈ S h , where S h is the finite element space. B Steven M. Wise

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Finite difference approximate solutions for the Cahn‐Hilliard equation

Cited by 33 publications

References 28 publications

Computationally efficient solution to the Cahn–Hilliard equation: Adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem

Computationally efficient solution to the Cahn–Hilliard equation: Adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem

A numerical method based on the boundary integral equation and dual reciprocity methods for one-dimensional Cahn–Hilliard equation

Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system

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