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

Dehghan, Mehdi; Mirzaei, Davoud

doi:10.1016/j.enganabound.2008.08.008

Cited by 37 publications

(19 citation statements)

References 61 publications

(92 reference statements)

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“…The same example was solved in [35,29] and the solutions obtained here are compared with these two previous works. Based on this example, convergence properties of the LS-SEM are discussed.…”

Section: Numerical Experimentsmentioning

confidence: 92%

“…This same problem was solved in [35,29] using a Fourier collocation method and a boundary integral based method respectively. In order to take the system (2) to a second order partial differential equation, we introduce the chemical potential defined by…”

Section: The Cahn-hilliard Equationmentioning

confidence: 99%

“…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%

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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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Section: Numerical Experimentsmentioning

confidence: 92%

Section: The Cahn-hilliard Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Fernandino

Dorao

2011

Applied Mathematical Modelling

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“…It has been documented that the solution provided by any numerical scheme that solves the Cahn-Hilliard equation must satisfy two main properties (De Mello and Silveira Filho, 2005;Dehghan and Mirzaei, 2009;Elliott and French, 1987;Novick-Cohen and Segel, 1984):…”

Section: The Cahn-hilliard Equationmentioning

confidence: 99%

“…Most of these theoretical results were validated computationally by several authors (see De Mello and Silveira Filho (2005); Eyre (1998); Lee et al (2011) among others). The numerical solution of entire cCH or CH equation have been investigated by various researchers using finite difference (Christlieb et al, 2012;Cueto-Felgueroso and Peraire, 2008;Du and Nicolaides, 1991;Eyre, 1998), finite elements French, 1987, 1989;Wells et al, 2006), boundary integral (Dehghan and Mirzaei, 2009), Fourier spectral (Zhu et al, 1999) methods -to name a few. In this work we consider finite difference schemes for both the cCH and CH equations.…”

Section: Introductionmentioning

confidence: 99%

On a fractional step-splitting scheme for the Cahn-Hilliard equation

Aderogba

Chapwanya

Djoko

2014

Engineering Computations

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For a partial differential equation with a fourth order derivative such as the CahnHilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. In this work, a fractional splitting method is employed to isolate the convective, the nonlinear second order and the fourth order differential terms. The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second order accuracy, we will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzburg-Landau energy and the coarsening properties of the solution.

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The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

Dehghan

Safarpoor

2015

Math Methods in App Sciences

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In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time-fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs . We employ the time-stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor-corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation.

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

Cited by 37 publications

References 61 publications

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

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

On a fractional step-splitting scheme for the Cahn-Hilliard equation

The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

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