A fourth-order spatial accurate and practically stable compact scheme for the Cahn–Hilliard equation

2014 ◽  
Vol 409 ◽  
pp. 17-28 ◽  
Author(s):  
Chaeyoung Lee ◽  
Darae Jeong ◽  
Jaemin Shin ◽  
Yibao Li ◽  
Junseok Kim
Keyword(s):  
Fourth Order ◽  
Compact Scheme ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

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

2014 ◽  
Vol 31 (7) ◽  
pp. 1151-1168 ◽  
Author(s):  
A.A. Aderogba ◽  
M. Chapwanya ◽  
J.K. Djoko
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Computational Time ◽  
Fractional Step ◽  
Step Method ◽  
Fractional Step Method ◽  
Order Accuracy ◽  
Content Type ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

Purpose – For a partial differential equation with a fourth-order derivative such as the Cahn-Hilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. The purpose of this paper is to employ a fractional splitting method to isolate the convective, the nonlinear second-order and the fourth-order differential terms. Design/methodology/approach – The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Findings – The scheme is second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Originality/value – The authors believe that this is the first time the equation is handled numerically using the fractional step method. Apart from the fact that the fractional step method substantially reduces computational time, it has the advantage of simplifying a complex process efficiently. This method permits the treatment of each segment of the original equation separately and piece them together, in a way that will be explained shortly, without destroying the properties of the equation.

High order approximations in space and time of a sixth order Cahn–Hilliard equation

2015 ◽  
Vol 30 (6) ◽  
Author(s):  
Oleg Boyarkin ◽  
Ronald H. W. Hoppe ◽  
Christopher Linsenmann
Keyword(s):  
Boundary Value Problem ◽  
Fourth Order ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Order Equation ◽  
Sixth Order ◽  
Fourth Order Equation ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractWe consider an initial-boundary value problem for a sixth order Cahn-Hilliard equation describing the formation of microemulsions. Based on a Ciarlet-Raviart type mixed formulation as a system consisting of a second order and a fourth order equation, the spatial discretization is done by a C

scholarly journals A fourth-order parabolic equation with a logarithmic nonlinearlity

2004 ◽  
Vol 69 (1) ◽  
pp. 35-48 ◽  
Author(s):  
Ahmed Bonfoh
Keyword(s):  
Free Energy ◽  
Parabolic Equation ◽  
Fourth Order ◽  
Constitutive Equations ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Hilliard Equation ◽  
Finite Dimensional ◽  
Fourth Order Parabolic Equation ◽  
Cahn Hilliard Equation

We consider some generalisations of the Cahn—Hilliard equation based on constitutive equations derived by M. Gurtin in (1996) with a logarithmic free energy. Compared to the classical Cahn—Hilliard equation (see [4, 5]), these models take into account the work of internal microforces and the anisotropy of the material. We obtain the existence and uniqueness of solutions results and then prove the existence of finite dimensional attractors.

Fourth-Order Spatial and Second-Order Temporal Accurate Compact Scheme for Cahn–Hilliard Equation

2019 ◽  
Vol 20 (2) ◽  
pp. 137-143
Author(s):  
Seunggyu Lee
Keyword(s):  
Fourth Order ◽  
Mass Conservation ◽  
Second Order ◽  
High Order Accuracy ◽  
Step Size ◽  
Order Accuracy ◽  
Central Difference ◽  
Hilliard Equation ◽  
Temporal And Spatial ◽  
Cahn Hilliard Equation

AbstractWe propose a fourth-order spatial and second-order temporal accurate and unconditionally stable compact finite-difference scheme for the Cahn–Hilliard equation. The proposed scheme has a higher-order accuracy in space than conventional central difference schemes even though both methods use a three-point stencil. Its compactness may be useful when applying the scheme to numerical implementation. In a temporal discretization, the secant-type algorithm, which is known as the second-order accurate scheme, is applied. Furthermore, the unique solvability regardless of the temporal and spatial step size, unconditionally gradient stability, and discrete mass conservation are proven. It guarantees that large temporal and spatial step sizes could be used with the high-order accuracy and the original properties of the CH equation. Then, numerical results are presented to confirm the efficiency and accuracy of the proposed scheme. The efficiency of the proposed scheme is better than other low order accurate stable schemes.

Error analysis of a finite element approximation of a degenerate Cahn-Hilliard equation

2018 ◽  
Vol 52 (3) ◽  
pp. 827-867 ◽  
Author(s):  
A. Agosti
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Fourth Order ◽  
Finite Element Approximation ◽  
Diffuse Interface ◽  
Degenerate Parabolic Equations ◽  
Element Approximation ◽  
Hilliard Equation ◽  
Degenerate Mobility ◽  
Cahn Hilliard Equation

This work considers a Cahn-Hilliard type equation with degenerate mobility and single-well potential of Lennard-Jones type, motivated by increasing interest in diffuse interface modelling of solid tumors. The degeneracy set of the mobility and the singularity set of the potential do not coincide, and the zero of the potential is an unstable equilibrium configuration. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. In particular, the singularities of the potential do not compensate the degeneracy of the mobility by constraining the solution to be strictly separated from the degeneracy values. The error analysis of a well posed continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality, is developed. Whilst in previous works the error analysis of suitable finite element approximations has been studied for second order degenerate and fourth order non degenerate parabolic equations, in this work the a priori estimates of the error between the discrete solution and the weak solution to which it converges are obtained for a degenerate fourth order parabolic equation. The theoretical error estimates obtained in the present case state that the norms of the approximation errors, calculated on the support of the solution in the proper functional spaces, are bounded by power laws of the discretization parameters with exponent 1/2, while in the case of the classical Cahn-Hilliard equation with constant mobility the exponent is 1. The estimates are finally succesfully validated by simulation results in one and two space dimensions.

scholarly journals On the sharp interface limit of a model for phase separation on biological membranes

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Abels ◽  
Johannes Kampmann
Keyword(s):  
Lipid Raft ◽  
Cell Membranes ◽  
System Modeling ◽  
Bulk Diffusion ◽  
Type Equation ◽  
Sharp Interface ◽  
Interface Problem ◽  
Hilliard Equation ◽  
System A ◽  
Cahn Hilliard Equation

AbstractWe rigorously prove the convergence of weak solutions to a model for lipid raft formation in cell membranes which was recently proposed in [H. Garcke, J. Kampmann, A. Rätz and M. Röger, A coupled surface-Cahn–Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci. 26 2016, 6, 1149–1189] to weak (varifold) solutions of the corresponding sharp-interface problem for a suitable subsequence. In the system a Cahn–Hilliard type equation on the boundary of a domain is coupled to a diffusion equation inside the domain. The proof builds on techniques developed in [X. Chen, Global asymptotic limit of solutions of the Cahn–Hilliard equation, J. Differential Geom. 44 1996, 2, 262–311] for the corresponding result for the Cahn–Hilliard equation.

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