Energy-stable linear schemes for polymer–solvent phase field models

Abstract

We present new linear energy-stable numerical schemes for numerical simulation of complex polymer-solvent mixtures. The mathematical model proposed by Zhou, Zhang and E (Physical Review E 73, 2006) consists of the Cahn-Hilliard equation which describes dynamics of the interface that separates polymer and solvent and the Oldroyd-B equations for the hydrodynamics of polymeric mixtures. The model is thermodynamically consistent and dissipates free energy. Our main goal in this paper is to derive numerical schemes for the polymersolvent mixture model that are energy dissipative and efficient in time. To this end we will propose several problem-suited time discretizations yielding linear schemes and discuss their properties. 45 generalization using the log-transformation of the elastic stress tensor and the Lagrange-type approximation of the convective term is possible.The purpose of the present paper is to derive energy-stable and runtimeefficient numerical schemes to solve the above-mentioned equations. This task has already been tackled by us in a preliminary fashion before [16], from which 50 paper we have also taken most of the wording of the present introduction. Compared to Reference [16], we have significantly improved the results, and also provide a much broader context and far more details.The paper is organized in the following way. In Section 2 we present a mathematical model for the polymer-solvent mixture consisting of the Cahn- 55Hilliard equation for the interface dynamics and the Oldroyd-B equations for the hydrodynamics. We also introduce a simplified model modelling only interface

Systematic derivation of hydrodynamic equations for viscoelastic phase separation

“…Fully in line with the arguments put forward in Ref. [17], we consider this as very important both from the point of view of fundamental physical consistency, but also from the point of view of mathematical analysis and derivation of stable and convergent numerical algorithms [34,35].…”

Relative energy estimates for the Cahn-Hilliard equation with concentration dependent mobility

“…We refer to [3,7,21] for results concerning the extension to multi-component systems and multiphysical problems. See also our recent works where logarithmic potentials and concentration dependent mobility functions have been used in the context of a complex model for viscoelastic phase separation [11,29,31]. Let us note that approximations in space and/or time and energy estimates are typically used to establish existence of solutions in rather general cases.…”

“…in the previous expression, where π0 τ : W h,τ → Q h,τ denotes the L 2 -orthogonal projection in time onto piecewise constants. Employing vh,τ = µ h,τ − μh,τ + r2,h,τ + απ 0 τ (φ h,τ − φh,τ ) ∈ Q h,τ as a test function in the identities (31) and (34), we further obtain…”

“…The first identity then follows by taking the difference of this equation with (31), and noting that…”

A Simple Benchmark Problem for the Numerical Methods of the Cahn–Hilliard Equation

“…where Ω ⊂ R d (d � 1, 2, 3) is a bounded domain, ϕ(x, t) is a compositional field, F(ϕ) � 0.25(ϕ 2 − 1) 2 , and ε is a positive constant. e CH equation was proposed for a model of the spinodal decomposition in a binary mixture and has been used to model many scientific phenomena such as topology optimization [4], phase separation [5][6][7], image processing [8], two-phase fluid flows [9,10], crystal model [11], tumor growth [12,13], and microstructure formations (see [14] for the basic principles and practical applications and [3] for the physical, mathematical, and numerical derivations of the CH equation).…”