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 of -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 -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.
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, 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 Section 2, 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
The above free energy can be rewritten as follows:with . The new free energy (2) is completely equal to the original energy (1). It should be noted that can balance the influence of nonlinear terms in numerical experiments, and it can also facilitate us to get estimate.
Taking the variational approach of the total free energy (2) in , one can get the following AC equation:subject to the initial and boundary condition
An important feature of the equation is that it satisfies the energy dissipation law. Taking the inner product of (3) with , we obtain
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
Given , we calculate as follows:
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 , which can be calculated by the following first-order Euler scheme: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:where
Proof. Computing the inner product of (6) with , and using the following identifiesThen, we findDivide 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:withThus, one can solve directly from (12).
One can obtain the weak form of (12): find , such thatThe above linear system can be rewritten aswith .
Next, we will show that linear system (15) has a unique solution. One can find thatwhere is a constant that depends on and . Moreover, note thatwhere depends on , and .
Furthermore, . Thus, the bilinear form 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 , and . This means that the bilinear form 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:where satisfies
Second, the pointwise error function can be denoted as
Subtracting (6) from (18) yieldswith
In order to obtain consistent results, the following lemmas will be used.
Lemma 2. There is a constant , such that
Then, we havewhere is a constant dependent on .
Proof. From the definition of , we findThus, for , we find
Lemma 3. Let be sequences of discrete function on . We find
Proof. We will apply mathematical induction to prove the above conclusion. For , we find . AssumeWhen , we findThis ends the proof.
To analyze the consistency results, we denote , such thatIt should be mentioned that Shen et al. [9] also made a similar assumption.
For simplicity of analysis, we set . The following lemma will show the 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 and bound in space direction). There is a constant ; if , we can get the uniform boundedness results as follows:
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, . It is easy to find that . Suppose that the numerical solution has an bound at :Then, we will check that is still valid. By the assumption of exact solution, we findTaking inner product of (21) with givesBy using Young’s inequality, we findUsing Lemma 2, we arrive atCombining (34)–(37), we obtainSumming up the above inequality for , we havewhereFor the first step, we note thatApplying discrete Gronwall’s inequality in (39) givesFrom (21), we haveSumming up for and using Lemma 3, we haveNote thatIf , we getThen, we obtain (31).
Theorem 2. Let be the solution of (4) and (5) and be the solution of (6). Under the assumption of Lemma 4, as , then the following error estimate holds:
Proof. Note that in Lemma 4, we prove the uniform boundedness results: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.
Set , and let us consider the following AC equation:
First, we will test the accuracy of time-discrete scheme by choosing a suitable source term such that the exact solution is
Let . The -error, -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 .
We calculate the errors by the following quantities.
The numerical results indicate that the time-discrete scheme is 2-order convergent in time direction. This is consistent with the theoretical results in Theorem 2.
Second, we fix , set in (49), and present the discrete energies of the time-discrete scheme for in Figures 1 and 2. Numerical results show that discrete energy decays, which is consistent with our proof.
In order to make a comparison with Choi et al. [11], we choose the same initial value as
Let . 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 .
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At last, the dynamic evolution of numerical solutions is also studied. Set , . 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.
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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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work of Shimin Lin is supported by National Natural Science Foundation of China (No. 11901237). The research of Fangying Song was supported by the NSFC (No. 11901100). The work of Tao Sun is supported in part by NSFC (No. 12171141). The work of Jun Zhang is supported by the National Natural Science Foundation of China (Nos. 11901132 and 62062018), Science and Technology Program of Guizhou Province (Nos. ZK[2022]006 and ZK[2022]031), Natural Science Research Projects of Education Department of Guizhou Province (No. KY[2021]015), and Guizhou Key Laboratory of Big Data Statistics Analysis (No. BDSA20200102).