scholarly journals The fast scalar auxiliary variable approach with unconditional energy stability for nonlocal Cahn–Hilliard equation

2020 ◽  
Vol 37 (1) ◽  
pp. 244-261
Author(s):  
Zhengguang Liu ◽  
Xiaoli Li
Keyword(s):  
Auxiliary Variable ◽  
Energy Stability ◽  
Hilliard Equation ◽  
Variable Approach ◽  
Cahn Hilliard Equation

scholarly journals Stability Condition of the Second-Order SSP-IMEX-RK Method for the Cahn–Hilliard Equation

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 11 ◽  
Author(s):  
Hyun Geun Lee
Keyword(s):  
Stability Condition ◽  
Second Order ◽  
Energy Stability ◽  
Order Method ◽  
Discrete Energy ◽  
Hilliard Equation ◽  
Strong Energy ◽  
Convex Splitting ◽  
Runge Kutta ◽  
Cahn Hilliard Equation

Strong-stability-preserving (SSP) implicit–explicit (IMEX) Runge–Kutta (RK) methods for the Cahn–Hilliard (CH) equation with a polynomial double-well free energy density were presented in a previous work, specifically H. Song’s “Energy SSP-IMEX Runge–Kutta Methods for the Cahn–Hilliard Equation” (2016). A linear convex splitting of the energy for the CH equation with an extra stabilizing term was used and the IMEX technique was combined with the SSP methods. And unconditional strong energy stability was proved only for the first-order methods. Here, we use a nonlinear convex splitting of the energy to remove the condition for the convexity of split energies and give a stability condition for the coefficients of the second-order method to preserve the discrete energy dissipation law. Along with a rigorous proof, numerical experiments are presented to demonstrate the accuracy and unconditional strong energy stability of the second-order method.

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.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

scholarly journals A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

2021 ◽  
pp. 110409
Author(s):  
Gerasimos Ntoukas ◽  
Juan Manzanero ◽  
Gonzalo Rubio ◽  
Eusebio Valero ◽  
Esteban Ferrer
Keyword(s):  
Free Energy ◽  
Discontinuous Galerkin ◽  
Hilliard Equation ◽  
Energy Stable ◽  
Cahn Hilliard Equation

The Cahn–Hilliard equation with a nonlinear source term

2021 ◽  
Vol 294 ◽  
pp. 88-117
Author(s):  
Alain Miranville
Keyword(s):  
Source Term ◽  
Nonlinear Source ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation
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