Maximum Principle Preserving Schemes for Binary Systems with Long-Range Interactions
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Maximum Principle Preserving Schemes for Binary Systems with Long-Range Interactions Xiang Xu1 · Yanxiang Zhao2 Received: 2 May 2020 / Revised: 10 July 2020 / Accepted: 18 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study some maximum principle preserving and energy stable schemes for the Allen– Cahn–Ohta–Kawasaki model with fixed volume constraint. With the inclusion of a nonlinear term f (φ) in the Ohta–Kawasaki free energy functional, we show that the Allen–Cahn– Ohta–Kawasaki dynamics is maximum principle preserving. We further design some first order energy stable numerical schemes which inherit the maximum principle preservation in both semi-discrete and fully-discrete levels. Furthermore, we apply the maximum principle preserving schemes to a general framework for binary systems with long-range interactions. We also present some numerical results to support our theoretical findings. Keywords Ohta–Kawasaki model · Gradient flow · Maximum principle preservation · Energy stability
1 Introduction Ohta–Kawasaki (OK) model is introduced in [1] and has been extensively applied for the study of phase separation of diblock copolymers, which have generated much interest in materials science in the past years due to their remarkable ability for self-assembly into nanoscale ordered structures [2]. Diblock copolymers are chain molecules made by two different segment species, say A and B species. Due to the chemical incompatibility, the two species tend to be phase-separated; on the other hand, the two species are connected by covalent chemical bonds, which leads to the so-called microphase separation. The OK model can describe such microphase separation for diblock copolymers via a free energy functional: E OK [φ] =
B
Td
1 γ |∇φ|2 + W (φ) dx + 2 2
Td
1
|(−)− 2 ( f (φ) − ω)|2 dx,
(1.1)
Yanxiang Zhao [email protected]
1
Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA
2
Department of Mathematics, George Washington University, Washington, DC 20052, USA 0123456789().: V,-vol
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with a volume constraint
Journal of Scientific Computing
(2020) 84:33
Td
( f (φ) − ω) dx = 0.
(1.2)
d Here Td = i=1 [−X i , X i ] ⊂ Rd , d = 2, 3 denotes a periodic box and 0 < 1 is an interface parameter that indicates the system is in deep segregation regime. φ = φ(x) is a phase field labeling function which represents the concentration of A species. By the assumption of incompressibility for the binary system, the concentration of B species can be implicitly represented by 1 − φ(x). The function W (φ) = 18(φ 2 − φ)2 is a double well potential which enforces the phase field function φ to be equal to 1 inside the interface and 0 outside the interface. Near the interfacial region, the phase field function φ rapidly but smoothly transitions from 0 to 1. A new term f (φ) = 3φ 2 − 2φ 3 is introduced in the free energy functional to resemble φ as the indicator for the A species. The first integral in (1.1) is
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