Unconditionally energy stable second-order numerical schemes for the Functionalized Cahn–Hilliard gradient flow equation based on the SAV approach

Abstract

In this paper, we devise and analyse three highly efficient second-order accurate (in time) schemes for solving the Functionalized Cahn–Hilliard (FCH) gradient flow equation where an asymmetric double-well potential function is considered. Based on the Scalar Auxiliary Variable (SAV) approach, we construct these schemes by splitting the FCH free energy in a novel and ingenious way. Utilizing the Crank–Nicolson formula, we firstly construct two semi-discrete second-order numerical schemes, which we denote by CN-SAV and CN-SAV-A, respectively. To be more specific, the CN-SAV scheme is constructed based on the fixed time step, while the CN-SAV-A scheme is a variable time step scheme. The BDF2-SAV scheme is another second-order scheme in which the fixed time step should be used. It is designed by applying the second-order backward difference (BDF2) formula. All the constructed schemes are proved to be unconditionally energy stable and uniquely solvable in theory. To the best of our knowledge, the CN-SAV-A scheme is the first unconditionally energy stable, second-order scheme with variable time steps for the FCH gradient flow equation. In addition, an effective adaptive time selection strategy introduced in Christlieb et al., (2014) is slightly modified and then adopted to select the time step for the CN-SAV-A scheme. Finally, several numerical experiments based on the Fourier pseudo-spectral method are carried out in two and three dimensions, respectively, to confirm the numerical accuracy and efficiency of the constructed schemes.

Introduction

Adding hydrophilic groups to the backbones of hydrophobic polymers, which is called functionalization, can make amphiphilic polymer materials. The Functionalized Cahn–Hilliard (FCH) free energy was proposed in [1] and improved in [2], [3], [4] to describe a binary mixture composed of the amphiphilic polymer and the polar solvent (such as water). In general, the FCH energy is non-increasing when the phase separation occurs in the mixture. Meanwhile, many different kinds of nanostructures gradually emerge, for example micelles, bilayers, pores, and so on. To study these interesting structures, the so-called FCH gradient flow equation was proposed, see [4]. This equation can be obtained by applying a dissipation mechanism on the FCH free energy. The FCH equation is a special case of the FCH gradient flow equation and can be formulated as

$${\varphi }_t=\Delta \mu ,$$$$\mu =({ϵ}^2\Delta -F^{\prime \prime }\left(\varphi \right)+{\eta }_1{ϵ}^p)({ϵ}^2\Delta \varphi -F^{\prime }\left(\varphi \right))+({\eta }_1-{\eta }_2){ϵ}^p{F}^{\prime }\left(\varphi \right),$$

where $ϵ\ll 1,{\eta }_1>0,{\eta }_2\in \mathbb{R}$ are constants, $\varphi$ is the phase-field variable and varies between $-1$ and $1$, and the double-well potential function $F\left(\varphi \right)$ is a nonlinear function. The FCH equation is actually an $H^{-1}$ gradient flow of the FCH free energy.

Generally speaking, it is almost impossible to obtain an analytic solution to the FCH equation, although a recent work [5] has established the global-in-time Gevrey regularity solutions for it. Hence, the approximate solution obtained by the numerical method becomes the focus. However, it is not easy to solve the FCH equation (1.1) numerically due to its high nonlinearity and high derivative order. In general, the explicit Euler method is one of the simplest methods to obtain the numerical solution. But for the FCH equation, a very small time step is necessary to ensure the numerical stability in calculations when this method is adopted. Hence, the explicit Euler method is not popular although it is very easy to implement. Therefore, designing some highly efficient numerical schemes for the FCH equation or the FCH gradient flow equation has become very important. In recent years, based on the ideas of highly efficient algorithms of Cahn–Hilliard-type equations, see [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], a number of numerical schemes have sprung up for approximating the solution to the FCH equation.

The fully implicit scheme was constructed and used in [20] to solve the FCH equation. Lots of numerical experiments have demonstrated that large time steps indeed can be used in this scheme, see [20], [21], [22]. However, a nonlinear system has to be solved, which may make the computation less efficient if an efficient iterative solver is not available. Hence, in order to improve the computational efficiency, both an adaptive time strategy and a highly efficient preconditioned iterative solver were designed by the authors. Additionally, higher order schemes of the FCH equation were also constructed in [23] by using the fully implicit method. However, the unconditional energy stability and unique solvability of all the above fully implicit schemes cannot be guaranteed, which greatly limits their applications.

Another classic method is Eyre’s convex-splitting method [24], which was introduced by Eyre for solving the Cahn–Hilliard equation. This method is widely used to design unconditionally energy stable schemes for phase field models, see [9], [12], [19]. A first-order scheme was proposed in [23] for the FCH equation based on the convex-splitting approach. This scheme satisfies the unconditional energy stability for Galerkin-type spatial discretization methods. However, its application is limited due to the lack of efficient iterative solvers. Another slightly modified version of this scheme, which we call implicit–explicit (IMEX) scheme in this paper, was also proposed in [23]. Unfortunately, the IMEX scheme is neither unconditionally energy stable nor uniquely solvable, although the decay in energy was verified by many numerical experiments. Based on the theory of convex-splitting, Feng and his co-workers established a novel first-order convex-splitting scheme by adding two auxiliary terms to the FCH energy [25]. The unconditional energy stability and unconditional unique solvability of the scheme were proved at the theoretical level. In addition, a second-order scheme based on the BDF2 formula was also presented in [25], but without any corresponding theoretical results. The main advantage of the convex-splitting method is that one can use it to design unconditionally energy stable schemes as long as a correct or reasonable splitting of the energy can be found. However, it should be noted that a nonlinear equation, in general, will be generated when the convex-splitting method is considered, which may make the calculation slow if an efficient iterative solver is unavailable.

The third approach is the so-called stabilization method that achieves great success in solving the classic Cahn–Hilliard equation [16] or other similar phase field equations [6], [11], [15], [18]. By adding stabilization terms to the semi-implicit scheme (i.e., evaluating the linear terms implicitly and the nonlinear terms explicitly), a linear, first-order, unconditionally energy stable scheme can be constructed. For the FCH equation, Chen and his co-workers put up a first-order scheme by using this method in [26]. Then, Guo and his co-workers adopted it and the local discontinuous Galerkin method to study the FCH equation in [27]. Unfortunately, neither of the two works gave the theoretical analysis of the energy stability (in time) of this time-semi-discrete scheme.

The Scalar Auxiliary Variable (SAV) method is very effective and easy to implement for solving the gradient flow equations [28], [29], [30], [31], [32]. One only needs to solve several linear, constant-coefficient equations in a SAV scheme. In [22], the SAV method was used for solving the FCH equation. However, large time steps are unallowable in the second-order scheme, although it is unconditionally energy stable. To solve this problem, a stabilization term was added to the second-order SAV scheme. It was shown in [22] that the new stabilized second-order scheme was unconditionally energy stable and unconditionally uniquely solvable. This work is remarkable and worthy of recognition. However, an extra linear transformation has to be considered for using this scheme when the potential function $F\left(\varphi \right)$ in (1.1a) is asymmetric. In addition, due to the introduction of the stabilization term, it is so hard, or almost impossible, to apply variable time steps to it without affecting the unconditional energy stability.

In this paper, we study the FCH gradient flow equation with various dissipation mechanisms, not just the FCH equation (1.1). Note that most of the above works focus on the study of the FCH equation. Based on the SAV approach, we successfully design three new second-order linear schemes for the FCH gradient flow equation (2.5) where an asymmetric double-well potential function $F\left(\varphi \right)$ is considered. The success is mainly due to the fact that a new auxiliary variable is introduced in a subtle way which is different from the one adopted in [22]. Specially, in order to greatly improve the computational efficiency, we construct a linear, second-order, unconditionally energy stable numerical scheme with variable time steps, which is almost impossible by using the splitting manner in [22]. We use the Fourier pseudo-spectral method [33] to discrete spatial variables in this paper when the periodic boundary conditions are considered. The Fourier pseudo-spectral method has a very low phase error and allows us to use fewer spatial nodes to achieve higher accuracy compared to other methods, for instance the finite element method [34], the finite volume method [35], [36] and the finite difference method [8], [25]. Moreover, by means of the Fast Fourier Transform (FFT) algorithm, the computational efficiency can be greatly improved. In addition, the Legendre spectral method [37] or the Chebyshev spectral method [38] can be used for non-periodic boundary conditions. However, it should be noted that the numerical solutions in this case may be polluted by the rounding-off errors resulting from approximations of high order derivatives explicitly.

The rest of the paper is organized as follows. In Section 2, we introduce the FCH free energy and the FCH gradient flow equation, respectively. In Section 3, we introduce the auxiliary variable so that the standard SAV method can be used to rewrite the original FCH gradient flow equation. In Section 4, we present the constructed second-order SAV schemes and establish the corresponding unconditional energy stability and unique solvability for each scheme. In Section 5, we perform some numerical experiments to verify the theoretical results. Finally, some conclusions follow in Section 6.