Unconditionally energy stable second-order numerical schemes for the Functionalized Cahn–Hilliard gradient flow equation based on the SAV approach

2021 ◽  
Vol 84 ◽  
pp. 16-38
Author(s):  
Chenhui Zhang ◽  
Jie Ouyang
Keyword(s):  
Gradient Flow ◽  
Flow Equation ◽  
Second Order ◽  
Numerical Schemes ◽  
Energy Stable

scholarly journals A Stabilized Fourier Spectral Method for the Fractional Cahn-Hilliard Equation

2018 ◽  
Vol 1 (3) ◽  
Author(s):  
Liquan Mei
Keyword(s):  
Spectral Method ◽  
Numerical Experiments ◽  
Second Order ◽  
Fourier Spectral Method ◽  
Numerical Schemes ◽  
Hilliard Equation ◽  
Discretization Schemes ◽  
Energy Stable ◽  
Cahn Hilliard Equation ◽  
Crank Nicolson

In this paper, the second order accurate (in time) energy stable numerical schemes are presented for the Fractional Cahn-Hilliard (CH) equation. Combining the stabilized technique, we apply the implicit Crank-Nicolson formula (CN) to derive second order temporal accuracy, and we use the Fourier spectral method for space discrete to obtain the fully discretization schemes. It is shown that the schemes are unconditionally energy stable. A few numerical experiments are presented to conclude the article.

scholarly journals Relating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials

2021 ◽  
Author(s):  
Alexander Mielke
Keyword(s):  
Gradient Flow ◽  
Careful Analysis ◽  
Flow Equation ◽  
Uniqueness Result ◽  
Gradient System ◽  
Independent System ◽  
Exact Relation ◽  
Flow Equations ◽  
Total Variation Flow ◽  
Energetic Solutions

AbstractWe consider a non-negative and one-homogeneous energy functional $${{\mathcal {J}}}$$ J on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-independent system given in terms of the time-dependent functional $${{\mathcal {E}}}(t,u)= t {{\mathcal {J}}}(u)$$ E ( t , u ) = t J ( u ) and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutions of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

Global Solutions to the Gradient Flow Equation of a Nonconvex Functional

2006 ◽  
Vol 37 (5) ◽  
pp. 1657-1687 ◽  
Author(s):  
G. Bellettini ◽  
M. Novaga ◽  
E. Paolini
Keyword(s):  
Gradient Flow ◽  
Global Solutions ◽  
Flow Equation ◽  
Nonconvex Functional
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