Effective Time Step Analysis of a Nonlinear Convex Splitting Scheme for the Cahn–Hilliard Equation

2019 ◽  
Vol 25 (2) ◽  
Author(s):  
Seunggyu Lee ◽  
Junseok Kim
Keyword(s):  
Splitting Scheme ◽  
Time Step ◽  
Effective Time ◽  
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.

Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Fengna Yan ◽  
Yan Xu
Keyword(s):  
Error Analysis ◽  
Discontinuous Galerkin ◽  
Implicit Time ◽  
Time Step ◽  
Discrete Scheme ◽  
Fully Discrete ◽  
Hilliard Equation ◽  
Local Discontinuous Galerkin ◽  
Energy Stable ◽  
Cahn Hilliard Equation

Abstract In this paper, we mainly study the error analysis of an unconditionally energy stable local discontinuous Galerkin (LDG) scheme for the Cahn–Hilliard equation with concentration-dependent mobility. The time discretization is based on the invariant energy quadratization (IEQ) method. The fully discrete scheme leads to a linear algebraic system to solve at each time step. The main difficulty in the error estimates is the lack of control on some jump terms at cell boundaries in the LDG discretization. Special treatments are needed for the initial condition and the non-constant mobility term of the Cahn–Hilliard equation. For the analysis of the non-constant mobility term, we take full advantage of the semi-implicit time-discrete method and bound some numerical variables in L ∞ L^{\infty} -norm by the mathematical induction method. The optimal error results are obtained for the fully discrete scheme.

Parameter-Free Time Adaptivity Based on Energy Evolution for the Cahn-Hilliard Equation

2016 ◽  
Vol 19 (5) ◽  
pp. 1542-1563 ◽  
Author(s):  
Fuesheng Luo ◽  
Tao Tang ◽  
Hehu Xie
Keyword(s):  
Large Time ◽  
Main Idea ◽  
Energy Evolution ◽  
Time Step ◽  
Time Stepping ◽  
Hilliard Equation ◽  
Phase Field Models ◽  
Adaptive Time ◽  
Cahn Hilliard Equation ◽  
Adaptive Time Stepping

AbstractIt is known that large time-stepping method are useful for simulating phase field models. In this work, an adaptive time-stepping strategy is proposed based on numerical energy stability and equi-distribution principle. The main idea is to use the energy variation as an indicator to update the time step, so that the resulting algorithm is free of user-defined parameters, which is different from several existing approaches. Some numerical experiments are presented to illustrate the effectiveness of the algorithms.

scholarly journals An Unsteady Model for Aircraft Icing Based on Tightly-Coupled Method and Phase-Field Method

Aerospace ◽  
2021 ◽  
Vol 8 (12) ◽  
pp. 373
Author(s):  
Hao Dai ◽  
Chengxiang Zhu ◽  
Ning Zhao ◽  
Chunling Zhu ◽  
Yufei Cai
Keyword(s):  
Numerical Simulation ◽  
Resistance Coefficient ◽  
Phase Field Method ◽  
Aircraft Icing ◽  
Shadow Zone ◽  
Step Method ◽  
Time Step ◽  
Hilliard Equation ◽  
Tightly Coupled ◽  
Cahn Hilliard Equation

An unsteady tightly-coupled icing model is established in this paper to solve the numerical simulation problem of unsteady aircraft icing. The multi-media fluid of air and droplets is regarded as a single medium fluid with variable material properties. Taking the droplet concentration as the phase parameter and the droplet resistance coefficient as the interphase force, the mass concentration distribution of the droplet is obtained by solving the Cahn–Hilliard equation. Fick’s law is introduced to improve the Cahn–Hilliard equation to predict the droplet shadow zone. On this basis, the procedure of the unsteady numerical simulation method for aircraft icing is established, including grid generation, the dual-time-step method to realize the unsteady calculation of the air and droplet tightly-coupled mixed flow field, and the improved shallow water icing model. Finally, through the comparative analysis of numerical examples, the effectiveness of the new model in predicting the droplet impact characteristics and the droplet shadow zone are verified. Compared with other icing models, the ice shapes predicted by the unsteady tightly-coupled model were found to be the most consistent with the experiments. In the icing comparison conditions in this manuscript, the prediction accuracy of the ice thickness at the stagnation point of the leading edge was up to 35% higher than that of LEWICE.

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