# A Novel Second-Order and Unconditionally Energy Stable Numerical Scheme for Allen–Cahn Equation

Authors: Shimin Lin; Fangying Song; Tao Sun; Jun Zhang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

We propose a novel time-stepping scheme for solving the Allen–Cahn equation. We first rewrite the free energy into an equivalent form and then obtain a new Allen–Cahn equation by energy variational formula ofL2-gradient flow. Using leapfrog formula, a new linear scheme is obtained, and we prove that the numerical scheme is unconditionally energy stable and uniquely solvable, and the discrete energy is in agreement with the original free energy. In addition, we also discuss the uniform boundedness and error estimate of numerical solution, the results show that the numerical solution is uniformly bounded in H2-norm, and error estimate shows that the time-stepping scheme can achieve second-order accuracy in time direction. At last, several numerical tests are illustrated to verify the theoretical results. The numerical strategy developed in this paper can be easily applied to other gradient flow models.


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## Body

## 1. Introduction

The Allen–Cahn(AC) equation is an important model for phase field simulation in materials science. It originally describes the phase transitions process of binary alloys at a certain temperature [1]. Its important applications can be found in the fields of image analysis, crystal growth, mean curvature flows, and so on [2–6]. As we all know, the AC equation can be derived from energy variational formula, that is, L2 gradient flow, which will produce a highly nonlinear function.In order to solve the AC equation more efficiently, two problems need to be solved. Firstly, the nonlinear term should be discretized appropriately. It is well known in numerical analysis that the implicit methods [7] have usually no time step restrictions, but they require the solution of a nonlinear system. On the other hand, the explicit time integrator [8, 9] does not require a solution of linear systems, but small step sizes are taken due to the stability restrictions. Secondly, the constructed numerical scheme is expected to be energy dissipative because this model is also energy dissipative in nature. Nowadays, many numerical methods are applied to solve AC equations. Hwang et al. [10] presented benchmark problems for the numerical methods of the phase field equations.In [11], Choi et al. proposed an unconditionally gradient stable scheme to solve the AC equation with a binary mixture. The pointwise boundedness of the numerical solution was obtained. Based on operator splitting techniques, Li et al. [12] presented a second-order hybrid numerical scheme to solve the AC equation with antiphase domain. They proved that the constructed numerical scheme can achieve second-order accuracy in time and space direction. Feng and Prohl [6] constructed some semi-discrete and fully discrete schemes for solving the AC equation. They also obtained some a priori error estimate results for the proposed numerical schemes.However, for the phase field model, it is still a challenge to construct a linear, second-order, unconditionally energy stable numerical scheme. The popular invariant energy quadratization (IEQ) [13–16] and scalar auxiliary variable (SAV) [17, 18] techniques can generate linear and unconditional energy numerical schemes. Based on the Runge–Kutta formula and SAV method, Akrivis et al. [19] constructed a linear and arbitrarily high-order numerical scheme for solving AC and Cahn–Hilliard equations. In fact, both IEQ and SAV methods need to assume that the nonlinear term is bounded from below. Moreover, the unconditional energy stability of the two methods is in agreement with the modified energy, but not the original free energy.In this work, a linear and second-order time-stepping scheme is developed to approximate the AC equation. Different from IEQ or SAV method, our new scheme can achieve unconditional energy stability without making any assumptions about nonlinear terms or introducing auxiliary functions. In addition, the discrete energy strictly corresponds to the original free energy. We also obtain the uniform boundedness of the numerical solution and prove the second-order accuracy of the numerical method in time direction. Finally, some numerical examples are performed to show the effectiveness of the numerical scheme.The rest of the article is organized as follows. In Section2, we will briefly introduce the constructed AC equation. In Section 3, unconditional stability, uniqueness, and convergence of the time-stepping scheme are studied in detail. In Section 4, some numerical experiments are performed to demonstrate the accuracy and unconditional stability of the time-stepping scheme. The conclusion remark of this paper will be given in the last section.


## 2. Energy Dissipation Property of AC Equations

Denote the total free energy(1)Eϕ=∫Ωε22∇ϕ2+ϕ2−12dx.The above free energy can be rewritten as follows:(2)Eϕ=∫Ωε22∇ϕ2+γϕ2+ϕ2−γ+222dx−γ24+γΩ,with γ≥0. The new free energy (2) is completely equal to the original energy (1). It should be noted that γϕ2 can balance the influence of nonlinear terms in numerical experiments, and it can also facilitate us to get L2 estimate.Taking the variational approach of the total free energy (2) in L2Ω, one can get the following AC equation:(3)ϕt−ε2Δϕ+4ϕ3−2γ+4ϕ+2γϕ=0,subject to the initial and boundary condition(4)ϕ|t=0=ϕ0,ϕisperiodic.An important feature of the equation is that it satisfies the energy dissipation law. Taking theL2 inner product of (3) with ϕt, we obtain(5)ddtEϕ=−ϕt2.


## 3. Allen–Cahn Equation

In this section, we will develop a second-order and linear discrete scheme for AC (3). The unconditional stability and uniqueness of time-discrete scheme are proved.


### 3.1. Unconditional Stability and Uniqueness of Time-Discrete Scheme

Givenϕn,ϕn−1, we calculate ϕn+1 as follows:(6)ϕn+1−ϕn−12δt−ε2Δϕn+1+ϕn−12+γϕn+1+ϕn−1+2ϕnϕn+1+ϕn−1ϕn−γ+2ϕn=0.Remark 1.
The above numerical scheme is different from IEQ or SAV method. We use the leapfrog formula to discretize the time direction and adopt the implicit-explicit method to deal with the nonlinear term. In order to obtain the energy stability, the IEQ and SAV methods need to assume that the free energy functions are bounded from below. But we can prove the unconditional stability of the numerical method without making any assumptions about the nonlinear term.Remark 2.
To initiate the second-order leapfrog scheme(6), we need the initial value ϕ1, which can be calculated by the following first-order Euler scheme:(7)ϕ1−ϕ0δt−ε2Δϕ1+2γϕ1+4ϕ03−2γ+4ϕ0=0.
For the above time-discrete scheme, we have the following energy stability results.Lemma 1.
The time-discrete scheme(6) satisfies the energy dissipation as follows:(8)Eϕn+1,ϕn=Eϕn,ϕn−1−14δtϕn+1−ϕn−12,where(9)Eϕn+1,ϕn=ε24∇ϕn+12+∇ϕn2+γ2ϕn+12+ϕn2+ϕn+1ϕn−γ+222−γ24+γΩ.Proof.
Computing the inner product of (6) with 2ϕn+1−ϕn−1, and using the following identifies(10)a2b+c−γ+2a×b−c=a2b2−γ+2ab+γ+222−a2c2−γ+2ac+γ+222,=ab−γ+222−ac−γ+222.
Then, we find(11)−1δtϕn+1−ϕn−12=ε2∇ϕn+12−∇ϕn−12+2γϕn+12−ϕn−12+4ϕn+1ϕn−γ+222−ϕnϕn−1−γ+222.
Divide both sides by 4, and we can derive (8).Theorem 1.
The time-discrete scheme (6) is uniquely solvable.Proof.
First, we can rewrite (6) as follows:(12)ϕn+1−ε2δtΔϕn+1+2γδtϕn+1+4δtϕn2ϕn+1=ζ1,with(13)ζ1=ϕn−1+ε2δtΔϕn−1−2γδtϕn−1−4δtϕn2ϕn−1+4γ+2δtϕn.
Thus, one can solveϕn+1 directly from (12).
One can obtain the weak form of (12): find ϕ∈H1Ω, such that(14)ϕ,φ+ε2δt∇ϕ,∇φ+2γδtϕ,φ+4δtϕnϕ,ϕnφ=ζ1,φ,φ∈H1Ω.
The above linear system can be rewritten as(15)Lϕ,φ=ζ1,φ,with L=1−ε2δtΔ+4δt+4δtϕn2.
Next, we will show that linear system (15) has a unique solution. One can find that(16)Lϕ,ϕ=ϕ2+ε2δt∇ϕ2+2γδtϕ2+4δtϕnϕ22≥c0ϕH12,where c0 is a constant that depends on ε and δt. Moreover, note that(17)Lϕ,φ≤c1ϕH1φH1,where c1 depends on ε,δt, and ϕnL∞.
Furthermore,Lϕ,φ=Lφ,ϕ. Thus, the bilinear form Lϕ,φ is coercive, bounded, and symmetric. Then, we conclude that linear system (15) admits a unique solution by using the Lax–Milgram theorem. Meanwhile, one may check that Lϕ,ϕ≥0, and Lϕ,ϕ=0↔ϕ≡0. This means that the bilinear form Lϕ,φ is positive definite.


### 3.2. Consistency and Convergence Analysis

Here, we will analyze the uniform boundedness and error estimate of numerical solutions. First, using Taylor expansion, one can derive the error equation as follows:(18)ϕ⋅,tn+1−ϕ⋅,tn−12δt−ε2Δϕ⋅,tn+1+ϕ⋅,tn−12+γϕ⋅,tn+1+ϕ⋅,tn−1+2ϕ⋅,tnϕ⋅,tn+1+ϕ⋅,tn−1ϕ⋅,tn−2+γϕ⋅,tn=r0n,where r0n satisfies(19)r0n≤Cδt2.Second, the pointwise error function can be denoted as(20)Eϕn=ϕ⋅,tn−ϕn,n=0,1,…,Tδt.Subtracting (6) from (18) yields(21)Eϕn+1−Eϕn−12δt−ε2ΔEϕn+1+Eϕn−12+γEϕn+1+Eϕn−1+χn=r0n,with(22)χn=2ϕ⋅,tnϕ⋅,tn+1+ϕ⋅,tn−1ϕ⋅,tn−2+γϕ⋅,tn−2ϕnϕn+1+ϕn−1ϕn−2+γϕn.In order to obtain consistent results, the following lemmas will be used.Lemma 2.
There is a constantC0>0, such that(23)maxn≤kϕ⋅,tnL∞,ϕnL∞≤C0.Then, we have(24)χn≤C1Eϕn+Eϕn+1+Eϕn−1,where C1>0 is a constant dependent on C0.Proof.
From the definition ofχn, we find(25)χn≤2ϕ2⋅,tnϕ⋅,tn+1−ϕn2ϕn+1+2ϕ2⋅,tnϕ⋅,tn−1−ϕn2ϕn−1+Eϕn≤2ϕ2⋅,tn−ϕn2ϕ⋅,tn+1+ϕn2Eϕn+1+2ϕ2⋅,tn−ϕn2ϕ⋅,tn−1+ϕn2Eϕn−1+Eϕn.
Thus, forn≤k, we find(26)χn≤C1Eϕn+Eϕn+1+Eϕn−1.Lemma 3.
Letφnn=0K−1 be sequences of discrete function on Ω. We find(27)φn+1≤∑m=1nφm+1+φm−1+φ0.Proof.
We will apply mathematical induction to prove the above conclusion. Forn=1, we find φ2−φ0≤φ2+φ0. Assume(28)φk−φ0≤∑m=1k−1φm+1+φm−1,forn≤k−1.
Whenn=k, we find(29)φk+1−φ0=φk+1−φk−1+φk−1−φ0≤∑m=1k−2φm+1+φm−1+φk+1−φk−1≤∑m=1kφm+1+φm−1.
This ends the proof.
To analyze the consistency results, we denoteρ, such that(30)ρ=max0≤t≤Tϕ⋅,tL∞+1.
It should be mentioned that Shen et al. [9] also made a similar assumption.
For simplicity of analysis, we setε=γ=1. The following lemma will show the L∞ uniform boundedness results of numerical solution.Lemma 4.
Assume the exact solution of the AC equation is smooth enough (at least 2-order differentiable in time andW2,∞ bound in space direction). There is a constant τ0>0; if δt<τ0, we can get the uniform boundedness results as follows:(31)ϕkL∞≤ρ,k=0,1,…,Tδt.Remark 3.
For the assumption of continuous solution of the AC equation, we can find some similar hypotheses in the following works [9, 13, 16].Proof.
In order to prove the above conclusion, mathematical induction will be used. For the first step,k=0. It is easy to find that ϕ0L∞≤ρ. Suppose that the numerical solution has an L∞ bound at tn:(32)ϕnL∞≤ρ,forn≤k.
Then, we will check thatϕk+1L∞≤ρ is still valid. By the assumption of exact solution, we find(33)ϕ⋅,tnL∞≤C˜,∇ϕ⋅,tnL∞≤C˜.
TakingL2 inner product of (21) with 2Eϕn+1−Eϕn−1 gives(34)1δtEϕn+1−Eϕn−12+∇Eϕn+12−∇Eϕn−12+2Eϕn+12−Eϕn−12=−2χn,Eϕn+1−Eϕn−1+2r0n,Eϕn+1−Eϕn−1.
By using Young’s inequality, we find(35)2r0n,Eϕn+1−Eϕn−1≤4δtr0n2+14δtEϕn+1−Eϕn−12,(36)−2χn,Eϕn+1−Eϕn−1≤4δtχn2+14δtEϕn+1−Eϕn−12.
Using Lemma2, we arrive at(37)χn≤CEϕn+Eϕn+1+Eϕn−1.
Combining (34)–(37), we obtain(38)∇Eϕn+12−∇Eϕn−12+2Eϕn+12−Eϕn−12+12δtEϕn+1−Eϕn−12≤Cδt5+CδtEϕn2+Eϕn+12+Eϕn−12.
Summing up the above inequality forn=1,…,k, we have(39)Ek+1¯+12δt∑n=1kEϕn+1−Eϕn−12≤E1¯+Cδt4+Cδt∑n=1kEn+1¯,where(40)Ek+1¯=∇Eϕn+12+∇Eϕn2+4Eϕn+12+Eϕn2.
For the first step, we note that(41)E1¯≤Cδt4.
Applying discrete Gronwall’s inequality in (39) gives(42)Ek+1¯+12δt∑n=1kEϕn+1−Eϕn−12≤Cδt4.
From (21), we have(43)ΔEϕn+1+Eϕn−1≤C1δtEϕn+1−Eϕn−1+Eϕn+1+Eϕn−1+χn+r0n.
Summing up forn=1,…,k and using Lemma 3, we have(44)ΔEϕk+1≤∑n=1kΔEϕn+1+Eϕn−1≤Cδt.
Note that(45)ϕk+1L∞≤Eϕn+1L∞+ϕ⋅,tn+1L∞≤CEϕn+1H11/2Eϕn+1H21/2+ϕ⋅,tn+1L∞≤Cδt3/2+ϕ⋅,tn+1L∞.
IfCδt3/2≤1, we get(46)ϕk+1L∞≤1+ϕ⋅,tn+1L∞≤ρ.
Then, we obtain (31).Theorem 2.
Letϕ be the solution of (4) and (5) and ϕkk=0K be the solution of (6). Under the assumption of Lemma 4, as δt⟶0, then the following error estimate holds:(47)ϕ⋅,tk−ϕkH1≤Cδt2,k=0,1,…,K.Proof.
Note that in Lemma4, we prove the uniform boundedness results:(48)ϕkL∞≤ρ,k=0,1,…,Tδt.
Following the process of (34)–(42), we can get the conclusion of the theorem.


## 3.1. Unconditional Stability and Uniqueness of Time-Discrete Scheme

Givenϕn,ϕn−1, we calculate ϕn+1 as follows:(6)ϕn+1−ϕn−12δt−ε2Δϕn+1+ϕn−12+γϕn+1+ϕn−1+2ϕnϕn+1+ϕn−1ϕn−γ+2ϕn=0.Remark 1.
The above numerical scheme is different from IEQ or SAV method. We use the leapfrog formula to discretize the time direction and adopt the implicit-explicit method to deal with the nonlinear term. In order to obtain the energy stability, the IEQ and SAV methods need to assume that the free energy functions are bounded from below. But we can prove the unconditional stability of the numerical method without making any assumptions about the nonlinear term.Remark 2.
To initiate the second-order leapfrog scheme(6), we need the initial value ϕ1, which can be calculated by the following first-order Euler scheme:(7)ϕ1−ϕ0δt−ε2Δϕ1+2γϕ1+4ϕ03−2γ+4ϕ0=0.
For the above time-discrete scheme, we have the following energy stability results.Lemma 1.
The time-discrete scheme(6) satisfies the energy dissipation as follows:(8)Eϕn+1,ϕn=Eϕn,ϕn−1−14δtϕn+1−ϕn−12,where(9)Eϕn+1,ϕn=ε24∇ϕn+12+∇ϕn2+γ2ϕn+12+ϕn2+ϕn+1ϕn−γ+222−γ24+γΩ.Proof.
Computing the inner product of (6) with 2ϕn+1−ϕn−1, and using the following identifies(10)a2b+c−γ+2a×b−c=a2b2−γ+2ab+γ+222−a2c2−γ+2ac+γ+222,=ab−γ+222−ac−γ+222.
Then, we find(11)−1δtϕn+1−ϕn−12=ε2∇ϕn+12−∇ϕn−12+2γϕn+12−ϕn−12+4ϕn+1ϕn−γ+222−ϕnϕn−1−γ+222.
Divide both sides by 4, and we can derive (8).Theorem 1.
The time-discrete scheme (6) is uniquely solvable.Proof.
First, we can rewrite (6) as follows:(12)ϕn+1−ε2δtΔϕn+1+2γδtϕn+1+4δtϕn2ϕn+1=ζ1,with(13)ζ1=ϕn−1+ε2δtΔϕn−1−2γδtϕn−1−4δtϕn2ϕn−1+4γ+2δtϕn.
Thus, one can solveϕn+1 directly from (12).
One can obtain the weak form of (12): find ϕ∈H1Ω, such that(14)ϕ,φ+ε2δt∇ϕ,∇φ+2γδtϕ,φ+4δtϕnϕ,ϕnφ=ζ1,φ,φ∈H1Ω.
The above linear system can be rewritten as(15)Lϕ,φ=ζ1,φ,with L=1−ε2δtΔ+4δt+4δtϕn2.
Next, we will show that linear system (15) has a unique solution. One can find that(16)Lϕ,ϕ=ϕ2+ε2δt∇ϕ2+2γδtϕ2+4δtϕnϕ22≥c0ϕH12,where c0 is a constant that depends on ε and δt. Moreover, note that(17)Lϕ,φ≤c1ϕH1φH1,where c1 depends on ε,δt, and ϕnL∞.
Furthermore,Lϕ,φ=Lφ,ϕ. Thus, the bilinear form Lϕ,φ is coercive, bounded, and symmetric. Then, we conclude that linear system (15) admits a unique solution by using the Lax–Milgram theorem. Meanwhile, one may check that Lϕ,ϕ≥0, and Lϕ,ϕ=0↔ϕ≡0. This means that the bilinear form Lϕ,φ is positive definite.


## 3.2. Consistency and Convergence Analysis

Here, we will analyze the uniform boundedness and error estimate of numerical solutions. First, using Taylor expansion, one can derive the error equation as follows:(18)ϕ⋅,tn+1−ϕ⋅,tn−12δt−ε2Δϕ⋅,tn+1+ϕ⋅,tn−12+γϕ⋅,tn+1+ϕ⋅,tn−1+2ϕ⋅,tnϕ⋅,tn+1+ϕ⋅,tn−1ϕ⋅,tn−2+γϕ⋅,tn=r0n,where r0n satisfies(19)r0n≤Cδt2.Second, the pointwise error function can be denoted as(20)Eϕn=ϕ⋅,tn−ϕn,n=0,1,…,Tδt.Subtracting (6) from (18) yields(21)Eϕn+1−Eϕn−12δt−ε2ΔEϕn+1+Eϕn−12+γEϕn+1+Eϕn−1+χn=r0n,with(22)χn=2ϕ⋅,tnϕ⋅,tn+1+ϕ⋅,tn−1ϕ⋅,tn−2+γϕ⋅,tn−2ϕnϕn+1+ϕn−1ϕn−2+γϕn.In order to obtain consistent results, the following lemmas will be used.Lemma 2.
There is a constantC0>0, such that(23)maxn≤kϕ⋅,tnL∞,ϕnL∞≤C0.Then, we have(24)χn≤C1Eϕn+Eϕn+1+Eϕn−1,where C1>0 is a constant dependent on C0.Proof.
From the definition ofχn, we find(25)χn≤2ϕ2⋅,tnϕ⋅,tn+1−ϕn2ϕn+1+2ϕ2⋅,tnϕ⋅,tn−1−ϕn2ϕn−1+Eϕn≤2ϕ2⋅,tn−ϕn2ϕ⋅,tn+1+ϕn2Eϕn+1+2ϕ2⋅,tn−ϕn2ϕ⋅,tn−1+ϕn2Eϕn−1+Eϕn.
Thus, forn≤k, we find(26)χn≤C1Eϕn+Eϕn+1+Eϕn−1.Lemma 3.
Letφnn=0K−1 be sequences of discrete function on Ω. We find(27)φn+1≤∑m=1nφm+1+φm−1+φ0.Proof.
We will apply mathematical induction to prove the above conclusion. Forn=1, we find φ2−φ0≤φ2+φ0. Assume(28)φk−φ0≤∑m=1k−1φm+1+φm−1,forn≤k−1.
Whenn=k, we find(29)φk+1−φ0=φk+1−φk−1+φk−1−φ0≤∑m=1k−2φm+1+φm−1+φk+1−φk−1≤∑m=1kφm+1+φm−1.
This ends the proof.
To analyze the consistency results, we denoteρ, such that(30)ρ=max0≤t≤Tϕ⋅,tL∞+1.
It should be mentioned that Shen et al. [9] also made a similar assumption.
For simplicity of analysis, we setε=γ=1. The following lemma will show the L∞ uniform boundedness results of numerical solution.Lemma 4.
Assume the exact solution of the AC equation is smooth enough (at least 2-order differentiable in time andW2,∞ bound in space direction). There is a constant τ0>0; if δt<τ0, we can get the uniform boundedness results as follows:(31)ϕkL∞≤ρ,k=0,1,…,Tδt.Remark 3.
For the assumption of continuous solution of the AC equation, we can find some similar hypotheses in the following works [9, 13, 16].Proof.
In order to prove the above conclusion, mathematical induction will be used. For the first step,k=0. It is easy to find that ϕ0L∞≤ρ. Suppose that the numerical solution has an L∞ bound at tn:(32)ϕnL∞≤ρ,forn≤k.
Then, we will check thatϕk+1L∞≤ρ is still valid. By the assumption of exact solution, we find(33)ϕ⋅,tnL∞≤C˜,∇ϕ⋅,tnL∞≤C˜.
TakingL2 inner product of (21) with 2Eϕn+1−Eϕn−1 gives(34)1δtEϕn+1−Eϕn−12+∇Eϕn+12−∇Eϕn−12+2Eϕn+12−Eϕn−12=−2χn,Eϕn+1−Eϕn−1+2r0n,Eϕn+1−Eϕn−1.
By using Young’s inequality, we find(35)2r0n,Eϕn+1−Eϕn−1≤4δtr0n2+14δtEϕn+1−Eϕn−12,(36)−2χn,Eϕn+1−Eϕn−1≤4δtχn2+14δtEϕn+1−Eϕn−12.
Using Lemma2, we arrive at(37)χn≤CEϕn+Eϕn+1+Eϕn−1.
Combining (34)–(37), we obtain(38)∇Eϕn+12−∇Eϕn−12+2Eϕn+12−Eϕn−12+12δtEϕn+1−Eϕn−12≤Cδt5+CδtEϕn2+Eϕn+12+Eϕn−12.
Summing up the above inequality forn=1,…,k, we have(39)Ek+1¯+12δt∑n=1kEϕn+1−Eϕn−12≤E1¯+Cδt4+Cδt∑n=1kEn+1¯,where(40)Ek+1¯=∇Eϕn+12+∇Eϕn2+4Eϕn+12+Eϕn2.
For the first step, we note that(41)E1¯≤Cδt4.
Applying discrete Gronwall’s inequality in (39) gives(42)Ek+1¯+12δt∑n=1kEϕn+1−Eϕn−12≤Cδt4.
From (21), we have(43)ΔEϕn+1+Eϕn−1≤C1δtEϕn+1−Eϕn−1+Eϕn+1+Eϕn−1+χn+r0n.
Summing up forn=1,…,k and using Lemma 3, we have(44)ΔEϕk+1≤∑n=1kΔEϕn+1+Eϕn−1≤Cδt.
Note that(45)ϕk+1L∞≤Eϕn+1L∞+ϕ⋅,tn+1L∞≤CEϕn+1H11/2Eϕn+1H21/2+ϕ⋅,tn+1L∞≤Cδt3/2+ϕ⋅,tn+1L∞.
IfCδt3/2≤1, we get(46)ϕk+1L∞≤1+ϕ⋅,tn+1L∞≤ρ.
Then, we obtain (31).Theorem 2.
Letϕ be the solution of (4) and (5) and ϕkk=0K be the solution of (6). Under the assumption of Lemma 4, as δt⟶0, then the following error estimate holds:(47)ϕ⋅,tk−ϕkH1≤Cδt2,k=0,1,…,K.Proof.
Note that in Lemma4, we prove the uniform boundedness results:(48)ϕkL∞≤ρ,k=0,1,…,Tδt.
Following the process of (34)–(42), we can get the conclusion of the theorem.


## 4. Numerical Experiments

In this section, we will propose several numerical examples to show the accuracy, convergence, and unconditional energy stability of the time-discrete scheme.Setγ=2, and let us consider the following AC equation:(49)ϕt−ε2Δϕ+4ϕ3−8ϕ+4ϕ=fx,y,t.First, we will test the accuracy of time-discrete scheme by choosing a suitable source termfx,y,t such that the exact solution is(50)ϕx,y,t=exp−tcosxcosy.LetT=1,ε=0.1. The L2-error, H1-error, and convergence order of time direction are presented in Table 1. The spatial discretization of the time-discrete scheme is handled by using the Fourier pseudo-spectral method with Fourier modes Nx=128,Ny=128.Table 1
Numerical error and convergence order for 2D AC equation.
δtL2-errorOrderH1-errorOrder0.16.6813E-02-2.8594E-01-0.022.9463E-031.93941.3413E-021.90090.017.1127E-042.05043.2606E-032.04040.0022.7408E-052.02321.2630E-042.01990.0016.8161E-062.00753.1430E-052.00660.00022.7147E-072.00261.2524E-062.00230.00016.7832E-082.00073.1296E-072.00060.000022.7109E-092.00051.2507E-082.0005We calculate the errors by the following quantities.(51)p1=log2uNn,2Δt−uN2n,Δt0uN2n,Δt−uN4n,Δt/20,p2=log2uNn,2Δt−uN2n,ΔtH1uN2n,Δt−uN4n,Δt/2H1.The numerical results indicate that the time-discrete scheme is 2-order convergent in time direction. This is consistent with the theoretical results in Theorem2.Second, we fixδt=0.001,Nx=Ny=128, set f=0 in (49), and present the discrete energies of the time-discrete scheme for ε=0.1,ε=0.25 in Figures 1 and 2. Numerical results show that discrete energy decays, which is consistent with our proof.Figure 1
The discrete energies for AC equation withε=0.1.Figure 2
The discrete energies for AC equation withε=0.05.In order to make a comparison with Choi et al. [11], we choose the same initial value as(52)ϕx,y,0=tanh0.25−x−0.52+y−0.52ε.Letε=0.035,dt=0.01,N=256,γ=1. Figures 3(a)–3(d) show the evolution process of initial concentration with time. We also observe that the circle gradually shrinks, which is consistent with the theoretical prediction [11]. We still choose the same parameters as above. Figures 4(a)–4(d) show the result behaviours of the proposed method with respect to ε.Figure 3
The evolution process of initial concentration with time: (a)t = 0; (b) t = 62; (c) t = 125; (d).
(a)(b)(c)(d)Figure 4
The result behaviours of the proposed method with respect toε: (a) ε=0.05; (b) ε=0.02; (c) ε=0.01; (d).
(a)(b)(c)(d)At last, the dynamic evolution of numerical solutions is also studied. Setϕ0=cosxcosy, δt=0.001,Nx=Ny=128. The snapshots of phase separation with different times are presented in Figure 5. This results show that our numerical method can capture the process of phase separation.Figure 5
Numerical solutions ofϕ for the AC equation using the full discrete scheme (6). Snapshots are taken at t=0,1,40,80,100,300, respectively.
(a)(b)(c)(d)(e)(f)


## 5. Conclusion

In this paper, by skillfully dealing with the nonlinear function, we proposed a novel second-order and linear scheme to solve AC equation. Uniqueness and unconditional energy stability of the numerical scheme are proved. Moreover, by constructing an appropriate auxiliary function, we prove the uniform boundedness of the numerical solution. Based on the uniform boundedness result, we get the error estimate of time direction. Finally, several numerical examples are presented to demonstrate the accuracy, stability, and efficiency of the numerical scheme, and the dynamic evolution of the AC equation is also discussed.



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