Conservative nonlinear difference scheme for the Cahn-Hilliard equation

Choo, Sang-Mok; Chung, S. K.; Kim, K.I.

doi:10.1016/s0898-1221(98)00171-0

Cited by 51 publications

(20 citation statements)

References 19 publications

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“…A finite difference scheme has been studied by Furihata et al [16] who have examined the boundedness of the solution. The error estimates have been obtained by Choo and Chung [4,5] and Choo et al [6] for conservative nonlinear difference schemes. Recently, Furihata [14] has proposed a difference scheme which inherits the decrease of the total energy.…”

Section: Introductionmentioning

confidence: 99%

A conservative difference scheme for the viscous Cahn–Hilliard equation with a nonconstant gradient energy coefficient

Choo

Chung

Lee

2004

Applied Numerical Mathematics

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Section: Introductionmentioning

confidence: 99%

A conservative difference scheme for the viscous Cahn–Hilliard equation with a nonconstant gradient energy coefficient

Choo

Chung

Lee

2004

Applied Numerical Mathematics

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“…Different numerical schemes to solve the C-H equation have been suggested in the literature. Among these are finite element approaches [16][17][18][19], finite difference schemes [20,21], discontinuous Galerkin techniques [22][23][24], Fourier spectral methods [25,26], collocation techniques [27], lattice Boltzmann method [28,6], boundary element method [29], multigrid techniques [30], among others. In addition, some adaptive grid techniques have also been presented [31,32].…”

Section: Introductionmentioning

confidence: 99%

The least squares spectral element method for the Cahn–Hilliard equation

Fernandino

Dorao

2011

Applied Mathematical Modelling

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a b s t r a c tThe problem of numerically resolving an interface separating two different components is a common problem in several scientific and engineering applications. One alternative is to use phase field or diffuse interface methods such as the Cahn-Hilliard (C-H) equation, which introduce a continuous transition region between the two bulk phases. Different numerical schemes to solve the C-H equation have been suggested in the literature. In this work, the least squares spectral element method (LS-SEM) is used to solve the CahnHilliard equation. The LS-SEM is combined with a time-space coupled formulation and a high order continuity approximation by employing C 11 p-version hierarchical interpolation functions both in space and time. A one-dimensional case of the Cahn-Hilliard equation is solved and the convergence properties of the presented method analyzed. The obtained solution is in accordance with previous results from the literature and the basic properties of the C-H equation (i.e. mass conservation and energy dissipation) are maintained. By using the LS-SEM, a symmetric positive definite problem is always obtained, making it possible to use highly efficient solvers for this kind of problems. The use of dynamic adjustment of number of elements and order of approximation gives the possibility of a dynamic meshing procedure for a better resolution in the areas close to interfaces.

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“…The outline is as follows: In Section 2, we consider a Crank-Nicolson scheme [10,11] which is uniquely solvable. In Section 3, we show the convergence of the approximate solution.…”

Section: Introductionmentioning

confidence: 99%

A second-order accurate difference scheme for an extended Fisher–Kolmogorov equation

Kadri

Omrani

2011

Computers & Mathematics with Applications

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a b s t r a c tIn this paper, we present a Crank-Nicolson type finite difference scheme to approximate the nonlinear evolutionary Extended Fisher-Kolmogorov (EFK) equation. We prove the existence of the solution by using the well-known Browder fixed-point theorem. The stability of this scheme is established in L ∞ -norm. The uniqueness and convergence of the solution are analyzed. We discuss an iterative algorithm for solving the difference scheme and prove its convergence.

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Conservative nonlinear difference scheme for the Cahn-Hilliard equation

Cited by 51 publications

References 19 publications

A conservative difference scheme for the viscous Cahn–Hilliard equation with a nonconstant gradient energy coefficient

A conservative difference scheme for the viscous Cahn–Hilliard equation with a nonconstant gradient energy coefficient

The least squares spectral element method for the Cahn–Hilliard equation

A second-order accurate difference scheme for an extended Fisher–Kolmogorov equation

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