A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation

2014 ◽  
Vol 128 (2) ◽  
pp. 377-406 ◽  
Author(s):  
Zhen Guan ◽  
Cheng Wang ◽  
Steven M Wise
Keyword(s):  
Splitting Scheme ◽  
Hilliard Equation ◽  
Convex Splitting ◽  
Cahn Hilliard Equation

scholarly journals Curve and Surface Smoothing Using a Modified Cahn-Hilliard Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Yongho Choi ◽  
Darea Jeong ◽  
Junseok Kim
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Computational Results ◽  
Fourier Spectral Method ◽  
Splitting Scheme ◽  
Surface Smoothing ◽  
Three Dimensional Objects ◽  
Hilliard Equation ◽  
Stable Splitting ◽  
Cahn Hilliard Equation

We present a new method using the modified Cahn-Hilliard (CH) equation for smoothing piecewise linear shapes of two- and three-dimensional objects. The CH equation has good smoothing dynamics and it is coupled with a fidelity term which keeps the original given data; that is, it does not produce significant shrinkage. The modified CH equation is discretized using a linearly stable splitting scheme in time and the resulting scheme is solved by using a Fourier spectral method. We present computational results for both curve and surface smoothing problems. The computational results demonstrate that the proposed algorithm is fast and efficient.

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.

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