scholarly journals An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

2019 ◽  
Vol 233 (1) ◽  
pp. 167-247 ◽  
Author(s):  
Chun Liu ◽  
Hao Wu
Keyword(s):  
Boundary Condition ◽  
Mathematical Analysis ◽  
Variational Approach ◽  
Dynamic Boundary Condition ◽  
Hilliard Equation ◽  
Condition Model ◽  
Dynamic Boundary ◽  
Boundary Condition Model ◽  
Model Derivation ◽  
Cahn Hilliard Equation

scholarly journals A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Makoto Okumura ◽  
Takeshi Fukao ◽  
Daisuke Furihata ◽  
Shuji Yoshikawa
Keyword(s):  
Boundary Condition ◽  
Numerical Solutions ◽  
Normal Derivative ◽  
Second Order ◽  
Dynamic Boundary Condition ◽  
Central Difference ◽  
Structure Preserving ◽  
Hilliard Equation ◽  
Dynamic Boundary ◽  
Cahn Hilliard Equation

<p style='text-indent:20px;'>We propose a structure-preserving finite difference scheme for the Cahn–Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM) proposed by Furihata and Matsuo [<xref ref-type="bibr" rid="b14">14</xref>]. In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme proposed by Fukao–Yoshikawa–Wada [<xref ref-type="bibr" rid="b13">13</xref>] is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for our proposed scheme. Computation examples demonstrate the effectiveness of our proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.</p>

Pion-pion amplitudes from the boundary condition model

1963 ◽  
Vol 4 (6) ◽  
pp. 327-330 ◽  
Author(s):  
W.W.S. Au ◽  
E.L. Lomon
Keyword(s):  
Boundary Condition ◽  
Condition Model ◽  
Boundary Condition Model
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