Fully-discrete Spectral-Galerkin numerical scheme with second-order time accuracy and unconditional energy stability for the anisotropic Cahn–Hilliard Model

Highlights

• We propose a full discrete spectral-Galerkin approach for solving the anisotropic Cahn–Hilliard model.

• The proposed schemes are second-order accurate, provably unconditionally energy stable, and easy to implement.

• All nonlinear terms are treated semi-explicitly, and only need to solve several fully decoupled linear equations at each time step.

Abstract

In this work, we construct a fully-discrete Spectral-Galerkin scheme for the anisotropic Cahn–Hilliard model. The scheme is based on the combination of a novel so-called explicit-Invariant Energy Quadratization method for time discretization and the Spectral-Galerkin approach for spatial discretization. The designed scheme only needs to solve several independent linear equations with constant coefficients at each time step, which demonstrates the high computational efficiency. The introduction of two auxiliary variables and the design of their associated auxiliary ODEs play a vital role in obtaining the linear structure and unconditional energy stability and thus avoiding the computation of a variable-coefficient system. The unconditional energy stability of the scheme is further rigorously proved, and the implementation process is given in detail. Through several 2D and 3D numerical simulations, we further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.

Introduction

In this paper, we aim to develop a fully-discrete Spectral-Galerkin scheme for the anisotropic phase-field model, which is formulated by coupling the Cahn–Hilliard equation with the anisotropic coefficient. This model has been used to describe the formation of multi-faceted pyramids on the surface of nano-scale crystals since the work in [1], [2]. Its idea of modeling is to introduce a scalar function to label the inside and outside the crystal surface. The total free energy of the system is originated from the isotropic Cahn–Hilliard model but the energy potential is multiplied with the anisotropic coefficient. The governing partial differential equation (PDE) is then derived by using the energetic variational method in the $H^{-1}$ space (called the Cahn–Hilliard dynamics) since the volume conservation property is also needed.

For the development of numerical algorithms, it is a difficult task to design effective algorithms (especially algorithms that are not only energy stable but also easy to implement) to solve the model although it has received long-term attention due to the existence of the anisotropic coefficient. To the best of the author’s knowledge, the currently available energy-stable schemes include the nonlinear schemes based on the convex-splitting method (cf. [3], [4]), and the linear schemes based on the IEQ (Invariant Energy Quadratization) method (cf. [5]) or SAV (Scalar Auxiliary Variable) method (cf. [6]). Remarkably, we also note that a semi-implicit type scheme with first-order time accuracy was developed in [7], however, the anisotropic coefficient in [7] is assumed to be homogeneous of degree one, which is quite different from the model we study in this article.

It is worth noting that from the perspective of time accuracy, the convex splitting type scheme designed in [3] is first-order time-accurate, while the IEQ/SAV type schemes designed in [5], [6] has second-order time accuracy. But for spatial discretization, the convex splitting type schemes developed in [3], [4] provide the fully-discrete version (where Fourier-Spectral method is used in [3], and the finite-difference method is used in [4]), while the IEQ/SAV method in [5], [6] only provides a time-discrete format where space is assumed to be continuous. Therefore, to summarize, as far as the author knows, the existing fully-discrete schemes to solve the anisotropic Cahn–Hilliard model proposed in [1], [2] only have the first-order accuracy in time [3], and no fully-discrete scheme can be claimed to own the following properties, that is, linear, second-order time accurate, and unconditional energy stable (for simplicity, we call a numerical scheme with these attributes as an “ideal” type scheme). Moreover, we also notice that in the simulation of phase-field type models [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], although the non-periodic Spectral method has been widely applied, little work is devoted to constructing a fully-discrete scheme based on it. This motivates us to focus on establishing an “ideal ” type fully-discrete Spectral-Galerkin scheme to efficiently solve the anisotropic Cahn–Hilliard phase field model in this article.

To this end, inspired by the IEQ method developed in [5], this article proposes a new explicit type splitting method for the anisotropic system. Because nonlinear terms are all discretized in the explicit manner, we call it the explicit-IEQ method to show its connection with the IEQ method given in [5]. In the new explicit-IEQ approach, the nonlinear potential is linearized by using the “quadratized” way, which is the same as the IEQ method. However, the IEQ method has to solve a linear system with variable coefficients, which in general can result in high computational costs. In contrast, the novel explicit-IEQ method not only can obtain a scheme with all the “ideal” characteristics but also can completely overcome this inherent shortcoming. That is, at each time step, the explicit-IEQ method only needs to solve a few completely independent linear equations with constant coefficients, and so it has very high efficiency in practice.

The key idea of this new explicit-IEQ method is to introduce two auxiliary variables and couple two specially designed ordinary differential equations (ODEs) into the original system. Specifically, one variable (local type) is used to quadratize the nonlinear energy potential, and the chemical potential is then reformulated into an equivalent form. Note that this process is the same as the IEQ method in [5]. The significant difference lies in the introduction of another nonlocal variable and its associated ODE, which contains the inner products of the nonlinear terms with certain functions. Through an ingenious way of coupling, these two variables and their ODEs are combined with the original PDE model to form a final equivalent system. The advantage of the new format of the model is that one can easily achieve unconditional energy stability by simply discretizing the nonlinear terms explicitly. Meanwhile, using the nonlocal variable, we can easily obtain the decoupling structure since each discrete equation can be decomposed into multiple sub-equations with only constant coefficients, which can be solved independently and efficiently. Moreover, the nonlocal auxiliary variable plays a vital role in avoiding the computation of a variable-coefficient system which is needed in the IEQ method of [5]. Consequently, at each time step, the constructed explicit-IEQ scheme only needs to solve several constant-coefficient linear elliptic equations, which illustrates the high computational efficiency. To the best of the author’s knowledge, the scheme constructed in this paper is the first fully-discrete scheme that can have the following “ideal” properties, that is, linear, decoupled, second-order time-accurate, constant-coefficient, and unconditionally energy stable for the anisotropic Cahn–Hilliard model.

The rest of the article is organized as follows. In Section 2, we briefly describe the anisotropic Cahn–Hilliard model and its energy dissipation structure. In Section 3, the fully-discrete Spectral-Galerkin numerical scheme is constructed for both of the linear/Willmore regularized anisotropic model and the detailed implementations are given. We also prove the solvability and unconditional energy stability of the proposed schemes rigorously, and carry out various 2D and 3D numerical tests to show its advantages in accuracy and stability. In Section 5, we give some concluding remarks.