A Second-Order Convex Splitting Scheme for a Cahn-Hilliard Equation with Variable Interfacial Parameters

2017 ◽  
Vol 35 (6) ◽  
pp. 693-710 ◽  
Author(s):  
Xiao Li
Keyword(s):  
Second Order ◽  
Splitting Scheme ◽  
Hilliard Equation ◽  
Convex Splitting ◽  
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 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.

A Second Order BDF Numerical Scheme with Variable Steps for the Cahn--Hilliard Equation

2019 ◽  
Vol 57 (1) ◽  
pp. 495-525 ◽  
Author(s):  
Wenbin Chen ◽  
Xiaoming Wang ◽  
Yue Yan ◽  
Zhuying Zhang
Keyword(s):  
Numerical Scheme ◽  
Second Order ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

scholarly journals Numerical Approximation of the Space Fractional Cahn-Hilliard Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Zhifeng Weng ◽  
Langyang Huang ◽  
Rong Wu
Keyword(s):  
Numerical Approximation ◽  
Second Order ◽  
Differentiation Formula ◽  
Numerical Examples ◽  
Backward Differentiation Formula ◽  
Hilliard Equation ◽  
Order Case ◽  
Energy Stable ◽  
Cahn Hilliard Equation ◽  
Fourier Spectral Methods

In this paper, a second-order accurate (in time) energy stable Fourier spectral scheme for the fractional-in-space Cahn-Hilliard (CH) equation is considered. The time is discretized by the implicit backward differentiation formula (BDF), along with a linear stabilized term which represents a second-order Douglas-Dupont-type regularization. The semidiscrete schemes are shown to be energy stable and to be mass conservative. Then we further use Fourier-spectral methods to discretize the space. Some numerical examples are included to testify the effectiveness of our proposed method. In addition, it shows that the fractional order controls the thickness and the lifetime of the interface, which is typically diffusive in integer order case.

Sign in / Sign up

Export Citation Format

Share Document