Numerical analysis for solving Allen-Cahn equation in 1D and 2D based on higher-order compact structure-preserving difference scheme

Abstract

In this paper, we present a fourth-order difference scheme for solving the Allen-Cahn equation in both 1D and 2D. The proposed scheme is described by the compact difference operators together with the additional stabilized term. As a matter of fact, the Allen-Cahn equation contains the nonlinear reaction term which is eminently proved that numerical schemes are mostly nonlinear. To solve the complexity of nonlinearity, the Crank-Nicolson/Adams-Bashforth method is applied in order to deal with the nonlinear terms with the linear implicit scheme. The well-known energy-decaying property of the equation is maintained by the proposed scheme in the discrete sense. Additionally, the $L_{\infty }$ error analysis is carried out in the 1D case in a rigorous way to show that the method is fourth-order and second-order accuracy for the spatial and temporal step sizes, respectively. Concurrently, we examine the $L_2$ and $H^1$ error analysis for the scheme in the case of 2D. We consider the impact of the additional stabilized term on numerical solutions. The consequences confirm that an appropriate value of the stabilized term yields a significant improvement. Moreover, relevant results are carried out in the numerical simulations to illustrate the faithfulness of the present method by the confirmation of existing pieces of evidence.

Introduction

The diffusive-interface phase field approach for modeling the musicale morphological pattern formation and interface motion has been the subject of extensive investigation in variety of physical phenomena in applied sciences and engineering applications. The Allen-Cahn equation is one of the basic model for the diffuse interface approach that is developed to study phase transitions and interfacial dynamic in material science [1], which was originally introduced by Allen and Cahn in [2] to describe the motion of anti-phase boundaries in crystalline solids. The model was proposed to replace the singular macroscopic treatment of a discontinuous surface with a smooth one. As a result, an interface can be induced by the singularities in the sharp interface description. Currently, the Allen-Cahn equation has been significantly applied to crystal growth [3], [4], [5], phase transitions [6], [7], image analysis [8], [9], [10], [11], grain growth [12], [13], [14], [15] and so forth. Due to its broad application, there is an increase in substantial interest in finding analytical solutions to this equation; however, the analytical solution to class of nonlinear Allen-Cahn equation has not yet been successfully achieved in general. Consequently, this signifies the necessity of studying numerical techniques to design robust and efficient numerical algorithms while putting emphasis on the explorations and experiments of the solution behaviors. Therefore, the development in numerical treatment for the Allen-Cahn equation is competitive and challenging in applied mathematics.

At this stage, the attempt is made to develop numerical treatment for solving the class of nonlinear Allen-Cahn equation in the following form

$$\frac{\partial u}{\partial t}={ϵ}^2\Delta u-f\left(u\right)$$

$${u|}_{t=0}=u_0\left(x\right),x\in \Omega$$

$${u|}_{\partial \Omega }=0,$$

where $\Omega \subset {\mathbb{R}}^d$ ($d=1,2$) is a compact domain. The small positive parameter $ϵ$, when compared to the characteristic length of the laboratory scale [16], is normally referred as the inter-facial width. The quantity $u$ is defined as the difference between the concentrations of the two components in a mixture. When the mixtures can not pass through the boundary walls, the homogeneous boundary condition is physically used. The nonlinear term $f\left(u\right)$ is the Helmholtz free energy density when the polynomial double-well potential forms $f\left(u\right)=u^3-u$. As a result, the Allen-Cahn equation is the $L_2-$gradient flow of the following Ginzburg-Landau free energy functional

$$E\left(u\right)={\int }_{\Omega }(\frac{{ϵ}^2}{2}{\left|\nabla u\right|}^2+F\left(u\right))d\Omega ,$$

where $F\left(u\right)=\frac{1}{4}{(u^2-1)}^2$. Additionally, we find that the solution of the Allen-Cahn equation satisfies the energy-decaying property as follow:

$$\frac{dE\left(u\right(t\left)\right)}{dt}=-{\int }_{\Omega }{\left|u_t\right|}^{2}d\Omega \le 0,$$

which suggests that the total energy is decreasing in time $E\left(u\left(t_2\right)\right)\le E\left(u\left(t_1\right)\right)$ when $t_1<t_2\in (0,T].$

It is eminent that the small positive parameter $ϵ$ and the nonlinear term are integrated with the Allen-Cahn equation which become numerically difficult to solve. In addition, numerical techniques are somewhat complicated due to different approaches and computational coding. Additionally, formulating numerical schemes that inherit curtain property at the discrete level and the so-called structure-preserving property can gain a lot of advantages in numerical simulations. In the modern scientific computing, various scientific evidences affirm that the structure-preserving scheme acquires better results since it can reduce numerical oscillations. For the Allen-Cahn equation, it is found that the solution satisfies the energy-decaying property, as discussed earlier; moreover, there is another intrinsic property that is the maximum principle. Accordingly, there are abundant researches devoted the study and the development of structure-preserving scheme, including energy-decaying and maximum-principle properties for solving the Allen-Cahn equation [17], [18], [19], [20], [21], [22], [23]. Feng et al. [17] proposed two time discretization schemes by combining the Crank-Nicolson/Adams-Bashforth method and the Galerkin method, and proved that the schemes are either unconditionally energy stable, or conditionally energy stable under certain stability conditions. Later, the general linear multistep implicit-explicit schemes with axillary parameters which inherit nonlinear stability property were introduced by Feng et al. [18]. The theoretical foundations for their schemes were also discussed. In [19], Tang and Yang introduced some extra perturbation term in the implicit-explicit discretization in time and central finite difference in space in order to obtain its unconditional stability and the discrete maximum principle preserving. According to the literature review, most of the existing methods usually approximate the linear terms implicitly and the full nonlinear term explicitly, consequently, conditional stability of the methods are considerably expected. A concept of an unconditionally gradient stable scheme was introduced and developed by Eyre [20], which is a first-order accurate time-stepping method of non-linear energy stable for phase field models. Later, Choi et al. [21] proposed an unconditionally gradient stable scheme for solving the Allen-Cahn equation and obtained the discrete maximum principle and energy decreasing properties. In [22], the energy stable method based on stabilized first-order and second-order semi-implicit schemes was also established. A class of extrapolated and linearized RungeKutta methods based on the scalar auxiliary variable formulation for the AllenCahn and CahnHilliard phase field equations was developed and analyzed by Akrivis et al. [23]. Related researches on the nonlinear parabolic equations including the AllenCahn equation are evidently referred in [24], [25], [26]. In addition, the class of approximate analytical methods is practically applied to solve the Allen-Cahn equation [27], [28], [29], [30]; however, these methods cause loss of rigorous error estimates and structure-preserving property.

Among numerical methods, finite difference method (FDM) is one of the most effective and widely-used numerical methods for solving not only the Allen-Cahn equation, but also the class of nonlinear differential equations (see [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]). FDM distinctively manifests extensiveness and flexibility in its numerical method as it is implemented and applied to the structure discretization assumptions of the original problems. Recently, there has been a growing interest in the computation of the Allen-Cahn equation in terms of creating the structure-preserving finite difference scheme (FDS). In [31], the fully discrete numerical schemes were proposed by applying the second order exponential time differencing and the quadrature based finite difference discretization in space for solving the nonlocal Allen-Cahn equation. The discrete maximum principle and the discrete energy stability were also discussed. Hou and Leng [32] proposed a structure-preserving stabilized second-order CrankNicolson/Adams- Bashforth scheme of the Allen-Cahn equation including both the discrete maximum principle and the discrete energy stability. The second-order two-step backward differentiation formula with nonuniform grids for the Allen-Cahn equation was presented by Liao et al. [33], whereas the scheme of the formula preserves the discrete maximum principle and the discrete energy stability under the time-step ratio restriction. A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations was investigated by Liao et al. [34]. Very recently, a maximum-principle preserving and unconditionally energy-stable linear second-order finite difference scheme for Allen-Cahn equations were studied by Feng et al. [35]. An explicit maximum principle preserving integrating factor Runge-Kutta schemes were presented by Zhang et al. [36] with second-order in space and higher-order in time. In relevance to a critical review described earlier, plenty of numerical technique is regularly to formulate accurate results with the exact ones but the computational cost does not take into account. Hence, the standard second-order methods might be less suitable because more grid points are required to gain the level of acceptable accuracy. To resolve such settings, the compact finite difference schemes are substantially applied due to their significant advantages in high-performance computing while using the same grid stencils as compared with the classical finite difference methods (see [42], [43], [44], [45], [46], [47], [48], [49], [50], [51]). To the authors best knowledge, there is a few reported works based on compact finite difference methods for solving the Allen-Cahn equation [40], [41]. Obviously, the schemes in [40], [41] require heavy iterative computations because both the schemes are nonlinear implicit. For this reason, the first aim of this paper is to design a linear implicit difference scheme for solving the Allen-Cahn equation in both 1D and 2D based on compact difference operators. However, the direct application of compact difference operators to the Allen-Cahn equation might effect of structure-preserving property. The important features of this process are that the proposed schemes preserve energy principle and high-order accuracy. Another challenging task is to analyze the convergence of the compact FDS theoretically and prove the convergence rates for nonlinear Allen-Cahn equation. Subsequently, the aim of this paper is to create a high-accuracy linearly implicit structure-preserving scheme, which is based on compact difference operator approaches with the additional stabilized term [32], [50], [51].

The remaining part of the paper is structured as follows. In the next section, we provide the details of high-order implicit FDS for the 1D Allen-Cahn equation based on a compact difference operator and the additional stabilized term. The discrete energy-decaying property of the proposed scheme is also proven. By applying priori estimates, we then prove that the difference scheme is uniquely solvable, and convergence of the numerical solutions with second-order in time and fourth-order in space in a sense of $L_{\infty }-$norm can be acquired. In Section 3, the numerical scheme is extended to two-dimensional space, and the discrete energy-decaying property are also obtained. Furthermore, in order to obtain the error analysis, the regularity condition related to the maximum principle property is needed to prove the convergence and stability of $L_2-$ and $H^1$ norms. Additionally, the error analysis in the $L_{\infty }-$norm is numerically examined. In Section 4, comprehensive numerical results are reported to support and to validate the theoretical analysis and to confirm the performance of the proposed scheme. In conclusion section, this endeavor is discussed in brief and concluded towards the end of the research paper.