Energy Stable Numerical Schemes for Ternary Cahn-Hilliard System
- PDF / 1,364,978 Bytes
- 36 Pages / 439.37 x 666.142 pts Page_size
- 92 Downloads / 165 Views
Energy Stable Numerical Schemes for Ternary Cahn-Hilliard System Wenbin Chen1 · Cheng Wang2 Steven M. Wise5
· Shufen Wang3 · Xiaoming Wang4 ·
Received: 8 March 2020 / Revised: 8 June 2020 / Accepted: 26 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We present and analyze a uniquely solvable and unconditionally energy stable numerical scheme for the ternary Cahn-Hilliard system, with a polynomial pattern nonlinear free energy expansion. One key difficulty is associated with presence of the three mass components, though a total mass constraint reduces this to two components. Another numerical challenge is to ensure the energy stability for the nonlinear energy functional in the mixed product form, which turns out to be non-convex, non-concave in the three-phase space. To overcome this subtle difficulty, we add a few auxiliary terms to make the combined energy functional convex in the three-phase space, and this, in turn, yields a convex-concave decomposition of the physical energy in the ternary system. Consequently, both the unique solvability and the unconditional energy stability of the proposed numerical scheme are established at a theoretical level. In addition, an optimal rate convergence analysis in the ∞ (0, T ; H N−1 ) ∩ 2 (0, T ; H N1 ) norm is provided, with Fourier pseudo-spectral discretization in space, which is the first such result in this field. To deal with the nonlinear implicit equations at each time step, we apply an efficient preconditioned steepest descent (PSD) algorithm. A second order accurate, modified BDF scheme is also discussed. A few numerical results are presented, which confirm the stability and accuracy of the proposed numerical scheme. Keywords Ternary Cahn-Hilliard system · Convexity analysis · Energy stability · Optimal rate convergence analysis · Fourier pseudo-spectral approximation · Partial and total spreading Mathematics Subject Classification 35K30 · 35K55 · 65K10 · 65M12 · 65M70
1 Introduction The Cahn-Hilliard (CH) flow, which models spinodal decomposition and phase separation in a binary fluid [2,14,15], is one of the most well-known gradient flows. In the CH family of models, sharp interfaces are replaced by narrow diffusive transition layers, which often leads
B
Cheng Wang [email protected]
Extended author information available on the last page of the article 0123456789().: V,-vol
123
27
Page 2 of 36
Journal of Scientific Computing
(2020) 84:27
to models that are simpler and more theoretically tractable than their sharp interface counterparts. Most existing works for the CH flow are based on the two-phase model, including both the isotropic and anisotropic ones [17,24,29,36,39,62], etc. Meanwhile, if three or even more phase components are involved in the physical system, an interaction of these components has to be taken into consideration; see the related works [3–7,10,12,13,37,38,65]. There have been many different choices of free energy for the ternary CH system. In this article, we focus on the following thr
Data Loading...
