On linear schemes for a Cahn–Hilliard diffuse interface model

Guillén-González, Francisco; Tierra, Giordano

doi:10.1016/j.jcp.2012.09.020

Cited by 178 publications

(134 citation statements)

References 27 publications

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“…In order to obtain linear schemes following the ideas of Eyre [29], the authors in [37] rewrite the potential part of the free energy using a parameter β ≥ 0, as follows:…”

Section: Linear Eyre Approximation (El1)mentioning

confidence: 99%

“…But in [37] it is shown how the introduction of too much numerical dissipation usually leads to wrong equilibrium solutions.…”

Section: Remark 317mentioning

confidence: 99%

“…These schemes are based on the introduction of a Lagrange multiplier in the formulation of the potential term. These ideas were extended by us in [37] where two new linear second order in time approximations of the potential term are derived for the Cahn-Hilliard system.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Methods for Solving the Cahn–Hilliard Equation and Its Applicability to Related Energy-Based Models

Tierra

Guillén-González

2014

Arch Computat Methods Eng

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In this paper, we review some numerical methods presented in the literature in the last years to approximate the Cahn-Hilliard equation. Our aim is to compare the main properties of each one of the approaches to try to determine which one we should choose depending on which are the crucial aspects when we approximate the equations. Among the properties that we consider desirable to control are the time accuracy order, energy-stability, unique solvability and the linearity or nonlinearity of the resulting systems. In particular, we concern about the iterative methods used to approximate the nonlinear schemes and the constraints that may arise on the physical and computational parameters.Furthermore, we present the connections of the Cahn-Hilliard equation with other physically motivated systems (not only phase field models) and we state how the ideas of efficient numerical schemes in one topic could be extended to other frameworks in a natural way.

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“…In order to obtain linear schemes following the ideas of Eyre [29], the authors in [37] rewrite the potential part of the free energy using a parameter β ≥ 0, as follows:…”

Section: Linear Eyre Approximation (El1)mentioning

confidence: 99%

“…But in [37] it is shown how the introduction of too much numerical dissipation usually leads to wrong equilibrium solutions.…”

Section: Remark 317mentioning

confidence: 99%

See 1 more Smart Citation

Numerical Methods for Solving the Cahn–Hilliard Equation and Its Applicability to Related Energy-Based Models

Tierra

Guillén-González

2014

Arch Computat Methods Eng

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“…The main ingredient for the presented partially decoupled scheme is the choice of the transport velocities u φ and u ωi,M (i = 1, ..., M ). Using the idea presented in [6], [7] and [8], we update u n by a discrete time integration of the force densities. Hence we are able to postpone the computation of u n+1 until the very end without losing stability.…”

Section: Discrete Schemementioning

confidence: 99%

On numerical schemes for phase‐field models for electrowetting with electrolyte solutions

Metzger

2015

Proc Appl Math and Mech

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We present an energy-stable, decoupled discrete scheme for a recent model (see [1]) supposed to describe electrokinetic phenomena in two-phase flow with general mass densities. This model couples momentum and Cahn-Hilliard type phase-field equations with Nernst-Planck equations for ion density evolution and an elliptic transmission problem for the electrostatic potential. The transport velocities in our scheme are based on the old velocity field updated via a discrete time integration of the force densities. This allows to split the equations into three blocks which can be treated sequentially: The phase-field equation, the equations for ion transport and electrostatic potential, and the Navier-Stokes type equations. By establishing a discrete counterpart of the continuous energy estimate, we are able to prove the stability of the scheme.

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“…Numerical experiments are our indispensable tools to investigate the solution to this equation. Previously proposed schemes for the CahnHilliard equation [7,8,[10][11][12][13][14]17,29,34,35,38] and other kinetics equations contain fourth order term [23,25,26,28,37] could be used as valuable references. The main contribution of this work is to develop two second-order energy stable numeircal schemes for the two-dimensional diffuse interface model with Peng-Robinson EOS of single component substance.…”

Section: Introductionmentioning

confidence: 99%

Stability and Convergence Analysis of Second-Order Schemes for a Diffuse Interface Model with Peng-Robinson Equation of State

Peng¹,

Qiao²,

Sun³

2017

JCM

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On linear schemes for a Cahn–Hilliard diffuse interface model

Cited by 178 publications

References 27 publications

Numerical Methods for Solving the Cahn–Hilliard Equation and Its Applicability to Related Energy-Based Models

Numerical Methods for Solving the Cahn–Hilliard Equation and Its Applicability to Related Energy-Based Models

On numerical schemes for phase‐field models for electrowetting with electrolyte solutions

Stability and Convergence Analysis of Second-Order Schemes for a Diffuse Interface Model with Peng-Robinson Equation of State

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