1 Introduction

Phase-field equations have arisen as a significant numerical framework in modeling and studying evolution of pattern formation in materials [1]. In general, phase-field models are driven by gradient flows for the governing total energy functionals [2, 3], e.g., the Ginzburg–Landau (GL) free energy functional:

$$ \mathcal{E} (\phi ) = \int _{\Omega } \biggl( \frac{F(\phi )}{\epsilon ^{2}} + \frac{1}{2} \vert \nabla \phi \vert ^{2} \biggr) \,d \mathbf{x}, $$
(1)

where Ω is a domain in \(\mathbb{R}^{d}\) (\(d=1,2,3\)), ϕ is the order parameter, \(F(\phi ) = \frac{1}{4} (\phi ^{2}-1)^{2}\) is the GL potential, and \(\epsilon >0\) is a constant. We assume the homogeneous Neumann boundary condition for ϕ: \(\nabla \phi \cdot {\mathbf{n}} = 0\) on ∂Ω, where n is normal to ∂Ω. The \(L^{2}\)-gradient flow for (1) is the Allen–Cahn (AC) equation [4].

Various computational methods have been proposed to solve the AC equation [5–16]. Shen and Yang [5] and Long et al. [6] presented a second-order semiimplicit method based on a second-order backward differentiation formula with a stabilizing term. Guan et al. [7] and Bu et al. [8] developed second-order schemes using the convex splitting (CS) method [17, 18] and the secant scheme [19]. In [9], Lee and Lee developed a second-order semianalytical Fourier spectral method. Guillén-González and G. Tierra [10] and Li et al. [11] presented a second-order semiimplicit Crank–Nicolson (CN) method. In [12], Shin et al. proposed a second-order CS method based on the implicit–explicit (IMEX) Runge–Kutta (RK) method [20]. In [13], Ji et al. developed CN formulas using the invariant energy quadratization idea [21] and the scalar auxiliary variable approach [22]. In [14], Zhang et al. proposed a second-order scheme using the stabilized scalar auxiliary variable method [23].

Recently, the AC equation with a high-order polynomial \(F_{p}(\phi ) = \frac{1}{4} (\phi ^{p} - 1)^{2}\), referred to as the hAC equation, was introduced to better represent the interfacial dynamics [24]. Here, p is an even integer.

For more background information about the hAC equation, let us consider the origin of the standard quartic polynomial free energy, \(F(\phi ) = \frac{1}{4} (\phi ^{2}-1)^{2}\). In the original derivation of the bulk free energy in the total free energy functional \(\mathcal{E}(\phi )\), the following logarithmic free energy was used:

$$ F(\phi ) = \frac{\theta }{2} \biggl[ (1+\phi ) \ln \biggl( \frac{1+\phi }{2} \biggr) + (1-\phi ) \ln \biggl(\frac{1-\phi }{2} \biggr) \biggr] + \frac{\theta _{c}}{2}\bigl(1-\phi ^{2}\bigr), $$

where θ and \(\theta _{c}\) are the absolute and the critical temperatures, respectively [25]. However, the logarithmic bulk free energy function has singularities at \(\phi = \pm 1\). Therefore, for computational efficiency, a quartic polynomial approximation \(F(\phi ) = \frac{1}{4}(\phi ^{2} - 1)^{2}\) has been used instead of the logarithmic potential. The AC equation has been successfully applied as a building block equation for modeling many scientifically and industrially important problems such as crystal growth, image segmentation, motion by mean curvature, tissue growth, volume repairing and smoothing, topology optimization, volume reconstruction from point cloud and slice data, and multiphase fluid flows [24]. However, the AC equation with the classical quartic polynomial function has a limitation in preserving structures. For example, as we will show that with a numerical experiment in the later section, if two separated components are close to each other, then they may merge with each other. To resolve this problem, we use the hAC equation which has a good structure preserving property. As shown in Fig. 1, the higher the order p, the larger the free energy barrier. This fact implies that once two different phases are separated, they remain separated, which is a good feature in modeling the dynamics of complex shapes.

Figure 1
figure 1

Plot of the free energy \(F_{p}(\phi ) = \frac{1}{4} (\phi ^{p}-1)^{2}\) with different orders: \(p=2, 4, 10\)

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Compared to the AC equation, the hAC equation is less studied numerically [24]. The main purpose of this article is to develop a second-order energy stable method for the hAC equation, which is based on the CS idea. For \(F_{p}(\phi )\), one can split into \(\frac{1}{4} \phi ^{2p} + \frac{1}{4}\) and \(-\frac{1}{2} \phi ^{p}\), however, the resulting method is highly nonlinear thus its numerical implementation is complicated. In this study, we propose a linear CS that can be applied to all p, by placing \(\frac{1}{4} (\phi ^{p} - 1)^{2}\) in the concave part. In addition, we combine the linear CS with a strong-stability-preserving (SSP) IMEX-RK method [26] to obtain temporally second-order accurate and unconditionally energy stable scheme.

The contents of this article are as follows. We present the linear CS scheme with an auxiliary term in Sect. 2. In Sect. 3, we propose the second-order CS method, prove unconditional energy stability, and describe the computational implementation. In Sect. 4, we present computational tests to demonstrate the performance of the method. Conclusions are given in Sect. 5.

2 Linear convex splitting with an auxiliary term

It is well known that the AC equation preserves the maximum principle [27]. Thus, like a common practice to consider the AC and Cahn–Hilliard equations with the truncated Ginzburg–Landau double-well potential [5, 28–32], we can replace \(\frac{1}{4} (\phi ^{p} - 1)^{2}\) by

$$ {\tilde{F}}_{p}(\phi ) = \textstyle\begin{cases} \frac{A}{2} \phi ^{2} - B \phi + C, & \phi > 1, \\ \frac{1}{4} (\phi ^{p} - 1)^{2}, & \phi \in [ -1, 1 ], \\ \frac{A}{2} \phi ^{2} + B \phi + C, & \phi < -1, \end{cases} $$

where \(A = \frac{p^{2}}{2}\), \(B = \frac{p^{2}}{2}\), and \(C = \frac{p^{2}}{4}\).

Remark 1

With the replacement of \(\frac{1}{4} (\phi ^{p} - 1)^{2}\) by \({\tilde{F}}_{p}(\phi )\), we have

$$ \max_{\phi \in {\mathbb{R}}} \bigl\vert {\tilde{F}}_{p}''( \phi ) \bigr\vert \leq A. $$

Therefore, the hAC equation can be rewritten as follows:

$$ \frac{\partial \phi }{\partial t} = - \biggl( \frac{{\tilde{F}}_{p}'(\phi )}{\epsilon ^{2}} - \Delta \phi \biggr). $$
(2)

Next, we consider the following splitting \(\mathcal{E} (\phi ) = \mathcal{E}_{c} (\phi ) - \mathcal{E}_{e} ( \phi )\):

$$ \mathcal{E}_{c} (\phi ) = \int _{\Omega } \biggl( \frac{\beta }{2\epsilon ^{2}}\phi ^{2} + \frac{1}{2} \vert \nabla \phi \vert ^{2} \biggr) \,d \mathbf{x},\qquad \mathcal{E}_{e} (\phi ) = \int _{\Omega }\frac{1}{\epsilon ^{2}} \biggl( \frac{\beta }{2} \phi ^{2} - { \tilde{F}}_{p}(\phi ) \biggr) \,d\mathbf{x}, $$
(3)

where \(\beta \geq 0\) is a constant.

Lemma 1

Both \(\mathcal{E}_{c} (\phi )\) and \(\mathcal{E}_{e} (\phi )\) in (3) are convex if \(\beta \geq A\).

Proof

For \(\mathcal{E}_{c} (\phi )\), we have

$$ \begin{aligned} \mathcal{E}_{c} (\phi + s\psi ) &= \mathcal{E}_{c} ( \phi ) + \int _{\Omega }\frac{\delta \mathcal{E}_{c} (\phi )}{\delta \phi } s\psi \,d\mathbf{x} + \int _{\Omega }\frac{\delta ^{2} \mathcal{E}_{c} (\phi )}{\delta \phi ^{2}} \frac{(s\psi )^{2}}{2} \,d \mathbf{x} + O\bigl(s^{3}\bigr) \\ &= \mathcal{E}_{c} (\phi ) + s \int _{\Omega } \biggl( \frac{\beta }{\epsilon ^{2}}\phi - \Delta \phi \biggr) \psi \,d{ \mathbf{x}} + \frac{s^{2}}{2} \int _{\Omega } \biggl( \frac{\beta }{\epsilon ^{2}}\psi ^{2} - \psi \Delta \psi \biggr) \,d{ \mathbf{x}} + O\bigl(s^{3}\bigr). \end{aligned} $$

Then, we obtain

$$ \frac{d^{2} \mathcal{E}_{c} (\phi + s\psi )}{ds^{2}} \bigg|_{s=0} = \int _{\Omega } \biggl( \frac{\beta }{\epsilon ^{2}}\psi ^{2} + \vert \nabla \psi \vert ^{2} \biggr) \,d\mathbf{x} \geq 0. $$

For \(\mathcal{E}_{e} (\phi )\), we get

$$ \mathcal{E}_{e} (\phi + s\psi ) = \mathcal{E}_{e} (\phi ) + s \int _{\Omega }\frac{1}{\epsilon ^{2}} \bigl( \beta \phi - { \tilde{F}}_{p}'( \phi ) \bigr) \psi \,d\mathbf{x} + \frac{s^{2}}{2} \int _{\Omega }\frac{1}{\epsilon ^{2}} \bigl(\beta -{ \tilde{F}}_{p}''(\phi )\bigr) \psi ^{2} \,d\mathbf{x} + O\bigl(s^{3}\bigr). $$

Then, we have

$$ \frac{d^{2} \mathcal{E}_{e} (\phi + s\psi )}{ds^{2}} \bigg|_{s=0} = \int _{\Omega }\frac{1}{\epsilon ^{2}} \bigl(\beta -{ \tilde{F}}_{p}''( \phi )\bigr) \psi ^{2} \,d\mathbf{x} \geq 0\quad \text{if } \beta \geq A. $$

Therefore, the convexity condition of \(\mathcal{E}_{c} (\phi )\) and \(\mathcal{E}_{e} (\phi )\) is satisfied if \(\beta \geq A\). □

Lemma 2

The convexity condition of \({\mathcal{E}}_{c}(\phi )\) and \({\mathcal{E}}_{e}(\phi )\) results in the following inequality:

$$ \begin{aligned} {\mathcal{E}}(\phi ) - {\mathcal{E}}(\psi ) &\leq \biggl( \frac{\delta {\mathcal{E}}_{c}(\phi )}{\delta \phi } - \frac{\delta {\mathcal{E}}_{e}(\psi )}{\delta \phi }, \phi - \psi \biggr)_{L^{2}} \\ &= \biggl( \frac{\beta }{\epsilon ^{2}}\phi - \Delta \phi - \frac{1}{\epsilon ^{2}} \bigl(\beta \psi - {\tilde{F}}_{p}'(\psi )\bigr), \phi - \psi \biggr)_{L^{2}}, \end{aligned} $$
(4)

where \(( \cdot , \cdot )_{L^{2}}\) denotes the \(L^{2}\)-inner product with respect to Ω.

Proof

For \({\mathcal{E}}_{c}(\phi )\), we have

$$ \begin{aligned} & \int _{\Omega } \biggl( \frac{\beta }{2\epsilon ^{2}} \phi ^{2} + \frac{1}{2} \vert \nabla \phi \vert ^{2} - \frac{\beta }{2\epsilon ^{2}}\psi ^{2} - \frac{1}{2} \vert \nabla \psi \vert ^{2} \biggr) \,d \mathbf{x} \\ &\quad = \biggl( \frac{\beta }{\epsilon ^{2}}\phi - \frac{\beta }{2\epsilon ^{2}} (\phi -\psi ) - \Delta \phi + \frac{1}{2} \Delta (\phi -\psi ), \phi -\psi \biggr)_{L^{2}} \\ &\quad \leq \biggl( \frac{\beta }{\epsilon ^{2}}\phi - \Delta \phi , \phi - \psi \biggr)_{L^{2}}. \end{aligned} $$
(5)

And we obtain for \({\mathcal{E}}_{e}(\phi )\) with \(\beta \geq A\),

$$ \begin{aligned} & \int _{\Omega }\frac{1}{\epsilon ^{2}} \biggl( \frac{\beta }{2} \phi ^{2} - {\tilde{F}}_{p}(\phi ) - \frac{\beta }{2} \psi ^{2} + {\tilde{F}}_{p}(\psi ) \biggr) \,d\mathbf{x} \\ &\quad = \frac{1}{\epsilon ^{2}} \bigl( \beta \psi - {\tilde{F}}_{p}'( \psi ), \phi -\psi \bigr)_{L^{2}} + \biggl( \frac{1}{2\epsilon ^{2}} \bigl( \beta - {\tilde{F}}_{p}''(\varphi ) \bigr), ( \phi -\psi )^{2} \biggr)_{L^{2}} \\ &\quad \geq \frac{1}{\epsilon ^{2}} \bigl( \beta \psi - {\tilde{F}}_{p}'( \psi ), \phi -\psi \bigr)_{L^{2}}, \end{aligned} $$
(6)

where φ is between ϕ and ψ. Then, subtracting (6) from (5) yields inequality (4). □

3 Unconditionally stable second-order scheme

Next, we propose a second-order CS scheme for the hAC equation by combining the linear CS (3) with the specially designed second-order (three-stage) SSP-IMEX-RK scheme:

$$ \begin{aligned} &\phi ^{(1)} = \phi ^{n} - \Delta t \biggl( \frac{\delta {\mathcal{E}}_{c}(\phi ^{(1)})}{\delta \phi } - \frac{\delta {\mathcal{E}}_{e}(\phi ^{n})}{\delta \phi } \biggr), \\ &\phi ^{(2)} = a_{10} \phi ^{n} + a_{11} \phi ^{(1)} - b_{1} \Delta t \biggl( \frac{\delta {\mathcal{E}}_{c}(\phi ^{(2)})}{\delta \phi } - \frac{\delta {\mathcal{E}}_{e}(\phi ^{(1)})}{\delta \phi } \biggr), \\ &\phi ^{n+1} = a_{20} \phi ^{n} + a_{21} \phi ^{(1)} + a_{22} \phi ^{(2)} - b_{2} \Delta t \biggl( \frac{\delta {\mathcal{E}}_{c}(\phi ^{n+1})}{\delta \phi } - \frac{\delta {\mathcal{E}}_{e}(\phi ^{(2)})}{\delta \phi } \biggr), \end{aligned} $$
(7)

where \(a_{10}\), \(a_{11}\), \(a_{20}\), \(a_{21}\), \(a_{22}\), \(b_{1}\), \(b_{2}\) satisfy the second-order conditions and the stability condition [26], and are as follows:

$$\begin{aligned}& a_{10}=-\frac{1}{2},\qquad a_{11}= \frac{3}{2}, \qquad a_{20}=-\frac{1}{2},\qquad a_{21}=\frac{5}{2},\qquad a_{22}=-1, \\& b_{1}=\frac{1}{2},\qquad b_{2}= \frac{1}{2}. \end{aligned}$$

Theorem 1

The method (7) with \(\beta \geq A\) is unconditionally energy stable, i.e., we have the following inequality:

$$ {\mathcal{E}}\bigl(\phi ^{n+1}\bigr) \leq {\mathcal{E}}\bigl(\phi ^{n}\bigr) $$

for any time step \(\Delta t > 0\).

Proof

Let \(\mu (\phi ,\psi ) = \frac{\delta {\mathcal{E}}_{c}(\phi )}{\delta \phi } - \frac{\delta {\mathcal{E}}_{e}(\psi )}{\delta \phi }\) for simplicity of notation. From the second and third steps of (7), we obtain

$$ \phi ^{(2)} - \phi ^{(1)} = - \frac{\Delta t}{2} \bigl( \mu \bigl(\phi ^{(2)}, \phi ^{(1)}\bigr) + \mu \bigl(\phi ^{(1)},\phi ^{n}\bigr) \bigr) $$

and

$$ \phi ^{n+1} - \phi ^{(2)} = - \frac{\Delta t}{2} \bigl( \mu \bigl(\phi ^{n+1}, \phi ^{(2)}\bigr) - 2\mu \bigl(\phi ^{(2)},\phi ^{(1)}\bigr) - \mu \bigl(\phi ^{(1)}, \phi ^{n}\bigr) \bigr). $$

Using Lemma 2, we obtain

$$ \begin{aligned} & {\mathcal{E}}\bigl(\phi ^{n+1}\bigr) - { \mathcal{E}}\bigl(\phi ^{n}\bigr) \\ &\quad = \bigl( {\mathcal{E}}\bigl(\phi ^{n+1}\bigr) - {\mathcal{E}}\bigl( \phi ^{(2)}\bigr) \bigr) + \bigl( {\mathcal{E}}\bigl(\phi ^{(2)}\bigr) - {\mathcal{E}}\bigl(\phi ^{(1)}\bigr) \bigr) + \bigl( {\mathcal{E}}\bigl(\phi ^{(1)}\bigr) - {\mathcal{E}}\bigl(\phi ^{n}\bigr) \bigr) \\ &\quad \leq -\frac{\Delta t}{2} \bigl( \bigl( \mu \bigl(\phi ^{n+1}, \phi ^{(2)}\bigr), \mu \bigl(\phi ^{n+1},\phi ^{(2)}\bigr) - 2\mu \bigl(\phi ^{(2)},\phi ^{(1)} \bigr) - \mu \bigl( \phi ^{(1)},\phi ^{n}\bigr) \bigr)_{L^{2}} \\ &\qquad {} + \bigl( \mu \bigl(\phi ^{(2)},\phi ^{(1)}\bigr), \mu \bigl(\phi ^{(2)}, \phi ^{(1)}\bigr) + \mu \bigl(\phi ^{(1)},\phi ^{n}\bigr) \bigr)_{L^{2}} + 2 \bigl( \mu \bigl(\phi ^{(1)},\phi ^{n}\bigr), \mu \bigl(\phi ^{(1)},\phi ^{n}\bigr) \bigr)_{L^{2}} \bigr) \\ &\quad = -\frac{\Delta t}{8} \int _{\Omega } \bigl( \bigl( 2\mu \bigl(\phi ^{n+1}, \phi ^{(2)}\bigr) - 2\mu \bigl(\phi ^{(2)},\phi ^{(1)}\bigr) - \mu \bigl(\phi ^{(1)}, \phi ^{n} \bigr) \bigr)^{2} + 7 \bigl( \mu \bigl(\phi ^{(1)},\phi ^{n}\bigr) \bigr)^{2} \bigr) \,d\mathbf{x} \\ &\quad \leq 0. \end{aligned} $$

This completes the proof. □

3.1 Numerical implementation

We can rewrite the first-step of (7) as

$$ ( I + \Delta t L ) \phi ^{(1)} = \phi ^{n} + \frac{\Delta t}{\epsilon ^{2}} \bigl( \beta \phi ^{n} - {\tilde{F}}_{p}' \bigl( \phi ^{n}\bigr) \bigr), $$

where \(L= \beta / \epsilon ^{2} - \Delta \). Then, we have

$$ \phi ^{(1)} = ( I + \Delta t L )^{-1} \biggl( \phi ^{n} + \frac{\Delta t}{\epsilon ^{2}} \bigl( \beta \phi ^{n} - { \tilde{F}}_{p}'\bigl( \phi ^{n}\bigr) \bigr) \biggr), $$

where we have used the zero Neumann boundary condition for ϕ. In the same way,

$$ \phi ^{(2)} = \biggl( I + \frac{\Delta t}{2} L \biggr)^{-1} \biggl( - \frac{1}{2} \phi ^{n} + \frac{3}{2} \phi ^{(1)} + \frac{\Delta t}{2\epsilon ^{2}} \bigl( \beta \phi ^{(1)} - { \tilde{F}}_{p}'\bigl(\phi ^{(1)}\bigr) \bigr) \biggr) $$

and

$$ \phi ^{n+1} = \biggl( I + \frac{\Delta t}{2} L \biggr)^{-1} \biggl( - \frac{1}{2} \phi ^{n} + \frac{5}{2} \phi ^{(1)} - \phi ^{(2)} + \frac{\Delta t}{2\epsilon ^{2}} \bigl( \beta \phi ^{(2)} - { \tilde{F}}_{p}' \bigl(\phi ^{(2)}\bigr) \bigr) \biggr). $$

The Fourier spectral method [33–38] with the discrete cosine transform in MATLAB is used.

4 Computational experiments

4.1 Convergence test

We investigate the performance of the proposed scheme with an initial condition

$$ \phi (x,0) = 0.2\cos (4\pi x)+0.4\cos (7\pi x) $$

on \(\Omega =[0,1]\). We set \(\epsilon =0.02\) and \(\beta =0\); and calculate \(\phi (x,t)\) for \(0 < t \le t_{f}=5\epsilon ^{2}\). To demonstrate spatial accuracy of the numerical solution, computations are done by changing the number of grid points \(16,24,\ldots ,256\). Figure 2 displays the relative \(l_{2}\)-errors of \(\phi (x,t_{f})\) with \(p=4,6,8,10\) for different numbers of grid points and time steps. The errors are calculated by comparing with the reference numerical solution using 512 grid points and \(\Delta t = t_{f}/2^{14}\). As we can observe in Fig. 2, the spatial convergence of the scheme under grid refinement is evident.

Figure 2
figure 2

Relative \(l_{2}\)-errors of \(\phi (x,t_{f})\) with different p for \(16,24,\ldots ,256\) grid points and \(\Delta t=t_{f}/2^{12}, t_{f}/2^{11}, \ldots , t_{f}/2^{7}\) with \(\epsilon =0.02\)

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Next, to calculate the temporal convergence rate, computations are done by fixing \(\Delta x = 1/128\) and changing \(\Delta t=t_{f}/2^{12}, t_{f}/2^{11}, \ldots , t_{f}/2^{7}\). We use the quadruply over-resolved computational result as the reference numerical solution. Figure 3(a) displays the relative \(l_{2}\)-errors of \(\phi (x,t_{f})\) with \(p=4,6,8,10\) for different time steps. The errors are calculated by comparing with the reference numerical solution. The result shows that the scheme is second-order accurate in time for all p. In addition, Fig. 3(b) shows the average CPU time consumed using the method with \(p=4,6,8,10\) for different Δt. The computational results indicates that the average CPU time is almost linear with respect to the number of steps and not affected by p.

Figure 3
figure 3

(a) Relative \(l_{2}\)-errors of \(\phi (x,t_{f})\) with different p for \(\Delta t=t_{f}/2^{12}, t_{f}/2^{11}, \ldots , t_{f}/2^{7}\) with \(\epsilon =0.02\) and \(\Delta x=\frac{1}{128}\). (b) CPU time versus time step with different p

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4.2 Energy stability test

To study the energy stability of the proposed method, let us consider the following initial condition on \(\Omega =[0,1]\times [0,1]\):

$$ \phi (x,y,0) = \operatorname{rand}(x,y), $$

where \(\operatorname{rand}(x,y)\) is a random number between \([-0.9, 0.9]\). We use \(\epsilon =0.005\), \(\beta = p^{2}/2\), and \(\Delta x=\Delta y=1/128\). Figure 4 displays the temporal evolution of the discrete energy with various Δt for \(p=4,6,8,10\). And the temporal evolution of the discrete energy with \(\Delta t = 2^{19} \epsilon ^{2} \approx 13\) for \(p=4,6,8,10\) is displayed in Fig. 5. For all p and large time steps, all the discrete total energies are temporally decreasing. Figure 6 displays the temporal evolution of \(\phi (x,y,t)\) with \(\Delta t=2^{4} \epsilon ^{2}\) for all p.

Figure 4
figure 4

Evolution of the energy with different time steps for different p

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Figure 5
figure 5

Evolution of the energy with \(\Delta t = 2^{19} \epsilon ^{2} \approx 13\) for different p

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Figure 6
figure 6

Evolution of \(\phi (x,y,t)\) with \(\epsilon =0.005\), \(\Delta x=\Delta y=\frac{1}{128}\), and \(\Delta t=2^{4} \epsilon ^{2}\)

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4.3 Motion by mean curvature

Suppose a radially symmetric initial condition is given as follows:

$$ \phi (x,y,0) = \tanh \biggl( \frac{R_{0}-\sqrt{(x-a)^{2}+(y-b)^{2}}}{\sqrt{2}\epsilon } \biggr), $$

which represents a circle centered at \((a,b)\) with an initial radius \(R_{0}\). It is well known that the solution of the AC equation follows the motion by mean curvature. Therefore, the radius \(R(t)\) of the interfacial circle (where \(\phi (x,y,t)=0\)) shrinks at the rate of the curvature of the circle:

$$ \frac{dR(t)}{dt} = -\frac{1}{R(t)}, \quad \text{i.e.,}\quad R(t) = \sqrt{R_{0}^{2} - 2t}. $$

To compare the original AC equation and the hAC equation for the mean curvature flow, the computational domain \(\Omega =[0,1]\times [0,1]\), \((a,b)=(0.5,0.5)\), \(R_{0}=0.35\), \(\epsilon =0.02\), \(\beta = 0\), \(\Delta x=\Delta y=1/128\), and \(\Delta t=10^{-5}\) are used. Figure 7(a) displays the evolution of the radius of the interfacial circle for \(p=2,4,6,8,10\). And the interfacial layer (from \(\phi =-0.95\) to \(\phi =0.95\)) at \(t=0.05\) for \(p=2\) and 10 is displayed in Fig. 7(b). The results suggest that the hAC equation can generate a sharper interface than the original AC equation.

Figure 7
figure 7

(a) Evolution of the radius of the interfacial circle for different p. (b) Interfacial layer (from \(\phi =-0.95\) to \(\phi =0.95\)) at \(t=0.05\) for \(p=2\) and 10

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4.4 Effect of p on the interfacial dynamics in 2D

To investigate the effect of p in the hAC equation on the interfacial dynamics in 2D, we consider two circles on \(\Omega =[0,2]\times [0,1]\) (see the first column of Fig. 8(a)). Their centers are \((0.667,0.5)\) and \((1.333,0.5)\), respectively, and radii are 0.3, and we set \(\phi (x,y,0)\) to 1 inside the circles and −1 otherwise. We choose \(\epsilon =0.02\), \(\beta = 0\), \(\Delta x=\Delta y=1/128\), and \(\Delta t=10^{-5}\). Figure 8 displays the evolution of \(\phi (x,y,t)\) for \(p=2\) and 10. The initial two circles merge into one when \(p=2\), whereas shrink without merging when \(p=10\).

Figure 8
figure 8

Evolution of \(\phi (x,y,t)\) with \(\epsilon =0.02\), \(\Delta x=\Delta y=\frac{1}{128}\), and \(\Delta t=10^{-5}\). In each case, times are \(t=0\), \(400\Delta t\), \(2000\Delta t\), and \(4000\Delta t\) (from left to right)

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4.5 Effect of p on the interfacial dynamics in 3D

To examine the effect of p in the hAC equation on the interfacial dynamics in 3D, we consider a three-dimensional spiral on \(\Omega =[0,1]\times [0,1]\times [0,1]\) (see the first column of Fig. 9) [24]. The width and height of the spiral are \(8\Delta x\) and \(58\Delta x\), respectively, and we set \(\phi (x,y,z,0)\) to 1 inside the spiral and −1 otherwise. We use \(\Delta x=\Delta y=\Delta z=\frac{1}{64}\), \(\epsilon =\Delta x\), \(\beta = p^{2}/2\), and \(\Delta t=10\epsilon ^{2}\). Figures 9 and 10 display the evolution of \(\phi (x,y,z,t)\) and energy for \(p=4,6,8,10\). As p increases, the high-order polynomial free energy term \(\int _{\Omega }\frac{F_{p}(\phi )}{\epsilon ^{2}} \,d\mathbf{x}\) influences the evolution of ϕ more strongly than the interfacial energy term \(\int _{\Omega }\frac{1}{2} |\nabla \phi |^{2} \,d\mathbf{x}\). As a result, ϕ loses its spiral shape by merge as p decreases, whereas ϕ still shows a spiral shape maintaining its interface as p increases.

Figure 9
figure 9

Evolution of \(\phi (x,y,z,t)\) with \(\Delta x=\Delta y=\Delta z=\frac{1}{64}\), \(\epsilon =\Delta x\), and \(\Delta t=10 \epsilon ^{2}\)

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Figure 10
figure 10

Evolution of the energy

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5 Conclusions

We developed a linear, second-order, and unconditionally energy stable method for the hAC equation. In order to handle \(\int _{\Omega }{\tilde{F}}_{p}(\phi ) \,d\mathbf{x}\), \(\frac{\beta }{2\epsilon ^{2}} \int _{\Omega }\phi ^{2} \,d\mathbf{x}\) is added, which yields the linear convex–concave decomposition of the total energy for \(\beta \geq A\). In addition, we combined the linear CS with the specially designed second-order SSP-IMEX-RK scheme. We demonstrated that the proposed method is efficient and unconditionally energy stable, and second-order temporally accurate. In addition, we confirmed that the hAC equation has different interfacial dynamics depending on the value of p.