scholarly journals Towards Infinite Tilings with Symmetric Boundaries

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 444
Author(s):  
Florian Stenger ◽  
Axel Voigt
Keyword(s):  
Finite Elements ◽  
Large Time ◽  
Time Discretization ◽  
Numerical Approach ◽  
Computational Effort ◽  
Special Boundaries ◽  
Hilliard Equation ◽  
Self Similar ◽  
Energy Stable ◽  
Cahn Hilliard Equation

Large-time coarsening and the associated scaling and statistically self-similar properties are used to construct infinite tilings. This is realized using a Cahn–Hilliard equation and special boundaries on each tile. Within a compromise between computational effort and the goal to reduce recurrences, an infinite tiling has been created and software which zooms in and out evolve forward and backward in time as well as traverse the infinite tiling horizontally and vertically. We also analyze the scaling behavior and the statistically self-similar properties and describe the numerical approach, which is based on finite elements and an energy-stable time discretization.

An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

2012 ◽  
Vol 11 (4) ◽  
pp. 1261-1278 ◽  
Author(s):  
Zhengru Zhang ◽  
Zhonghua Qiao
Keyword(s):  
Difference Scheme ◽  
Numerical Simulations ◽  
Large Time ◽  
Second Order ◽  
Time Stepping ◽  
Hilliard Equation ◽  
Adaptive Time ◽  
Energy Stable ◽  
Cahn Hilliard Equation ◽  
Adaptive Time Stepping

AbstractThis paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon. The numerical simulation of the Cahn-Hilliard model needs very long time to reach the steady state, and therefore large time-stepping methods become useful. The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations. The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time. The proposed scheme is proved to be unconditionally energy stable and mass-conservative. An error estimate for the numerical solution is also obtained with second order in both space and time. By using this energy stable scheme, an adaptive time-stepping strategy is proposed, which selects time steps adaptively based on the variation of the free energy against time. The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

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

A Uniquely Solvable, Energy Stable Numerical Scheme for the Functionalized Cahn–Hilliard Equation and Its Convergence Analysis

2018 ◽  
Vol 76 (3) ◽  
pp. 1938-1967 ◽  
Author(s):  
Wenqiang Feng ◽  
Zhen Guan ◽  
John Lowengrub ◽  
Cheng Wang ◽  
Steven M. Wise ◽  
...  
Keyword(s):  
Convergence Analysis ◽  
Numerical Scheme ◽  
Hilliard Equation ◽  
Energy Stable ◽  
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.

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