A Simple Benchmark Problem for the Numerical Methods of the Cahn–Hilliard Equation

We present a very simple benchmark problem for the numerical methods of the Cahn–Hilliard (CH) equation. For the benchmark problem, we consider a cosine function as the initial condition.,e periodic sinusoidal profile satisfies both the homogeneous and periodic boundary conditions. ,e strength of the proposed problem is that it is simpler than the previous works. For the benchmark numerical solution of the CH equation, we use a fourth-order Runge–Kutta method (RK4) for the temporal integration and a centered finite difference scheme for the spatial differential operator. Using the proposed benchmark problem solution, we perform the convergence tests for an unconditionally gradient stable scheme via linear convex splitting proposed by Eyre and the Crank–Nicolson scheme. We obtain the expected convergence rates in time for the numerical schemes for the one-, two-, and three-dimensional CH equations.

ere have been many research papers regarding the accurate numerical methods for the CH equation [15][16][17][18] (see the references therein for more details). In particular, a comparison study among various numerical methods for the CH equation was conducted on the perspective of stability and efficiency [19]. A test set developed to verify models, algorithms, or the accuracy of numerical methods is called "benchmark problem" [20]. However, there are only few benchmark problems for validating the accuracy of the proposed numerical methods. Recently, the authors in [21] proposed a benchmark problem for the two-and threedimensional CH equations. e benchmark problems were the shrinking annulus and spherical shell in the two-and three-dimensional cases, respectively. ey used the explicit Euler scheme with a very fine time step size, Δt < 10 −5 . e authors in [22] presented four benchmark problems for the Allen-Cahn (AC) and CH equations. e benchmark is the time T at which the value at a point in the domain changes from negative to positive. e authors in [20] presented two benchmark problems for phase-field models of solute diffusion and phase separation. Recently, a verification method for the convergence rates of the numerical solutions for wellknown parabolic partial differential equations was proposed in [23].
Typically, benchmark testing is essential for comparing the performance of a method. However, many researchers conduct self-tests using their own methods in most cases.
ere is a problem such that an error may occur relatively compared to the actual solution because an approximated solution converges to a numerical solution via an employed numerical method although convergence of a solution through numerical analysis is guaranteed. Moreover, most of these testing schemes are often implicit methods, which have relatively low accuracy compared to the results of long time simulation of explicit methods with a small time step. erefore, the main purpose of this paper is to present a very simple benchmark problem for the numerical methods of the CH equation, which does not employ the self-test but a classical explicit method for the temporal discretization. e strength of the proposed problem is that it is simpler than the previous works [20][21][22]. Furthermore, the benchmark problem can be applied to both Neumann and periodic boundary conditions. e contents of this paper are summarized as follows. In Section 2, a numerical solution is presented. In Section 3, numerical results are shown. Conclusions are made in Section 4.

Numerical Solution
We consider two discretization schemes to the CH equation for benchmark problem; one is the linearly stabilized splitting scheme (LSS) proposed by Eyre [24] which is firstorder accurate in time and the other is the Crank-Nicolson (CN) scheme [25] which is second-order in time. In order to perform a benchmark test for these two methods, we first need to obtain a benchmark numerical solution. To obtain the benchmark solution, we consider the one-dimensional CH equation: with the homogeneous Neumann boundary condition We define ϕ n i as an approximation of ϕ(x i , t n ). Now, we discretize equation (2) in time using a fourth-order Runge-Kutta method (RK4) [26] as follows: where

Convergence and Stability
Tests. First, we conduct a traditional convergence test. As a concrete example, an initial condition is given by Note that equation (4) can be used in testing numerical schemes with both the Neumann and periodic boundary conditions. We employ ε � 16h/(2 � 2 √ tanh −1 (0.9)) ≈ 0.03, which is the approximate transition layer width [27]. Table 1 shows the rate of convergence. We measure the accuracy of time at the final time T using ‖e Δt,0.5Δt ‖ 2 which is defined as follows: where e Δt,Δt/2 i is an error between the computed solutions using large and small time steps, respectively. More precisely, it is e Δt,Δt/2 In this test, we adopt the space step size as h � 1/10 and the final time as T � 10h 4 . We can confirm that RK4 is of fourth-order in time indeed according to Table 1.
For the next test, we investigate the maximum time step sizes Δt c with respect to the spatial step size h. Here, we fix N t � 10 5 and other parameters and the initial condition are the same as those described above. Table 2 lists the largest time step sizes satisfying that the numerical solution does not blow up. Figure 1 depicts the practically stable region and maximum time step size Δt c as described in Table 2. e explicit scheme is theoretically Δt ≈ O(h 4 ).

2
Discrete Dynamics in Nature and Society Figure 2 represents the CPU times (seconds) with respect to the number of spatial step N x , performed on Intel Core i5-6400 CPU at 2.70 GHz with 4 GB RAM. When we vary N x as 16, 32, 64, and 128 with fixed parameters, Δt � (1/128) 4 and N t � 10 7 , the CPU times are 88.812, 175.468, 354.14, and 708.718. at is, when N x is doubled, the CPU time is approximately doubled.

Benchmark Problem.
Let us consider the benchmark problem for the CH equation. We adopt equation (4)  tanh −1 (0.9)) ≈ 0.045. Figure 3 shows the initial condition and the reference numerical solution at t � 10 7 h 4 .

One-Dimensional
where Δt is a time step. We solve equations (7) and (8) using a multigrid method [25,28]. Note that we use the Gauss-Seidel iterative method in multigrid process combined with Newton's approximation to compute the nonlinear term in equation (8). Figures 4(a) and 4(b) show the errors for the LSS and CN scheme, respectively. Now, we define the discrete l 2 -norm and the discrete maximum norm as respectively. Tables 3 and 4 list the numerical errors and convergence rates of the LSS and those of the CN scheme at t � 10 7 h 4 , respectively. We confirm that both the schemes achieve the expected convergence rates (the first-order for the LSS and the second-order for the CN scheme).

Two-Dimensional
Space. Next, we consider the twodimensional version of equations (7) and (8) to the CH equation on Ω � (0, 1) × (0, 1). Straightforward extensions of the LSS and CN schemes are as follows: where for i � 1, . . . , N x and j � 1, . . . , N y . An initial condition is given by a simple extension of equation (4): We can minimize the influence of domain in this way. erefore, the two-dimensional reference numerical solution is extended from that of the one-dimensional as follows: Figures 5(a) and 5(b) show the initial condition and the reference solution at t � 10 7 h 4 , respectively.
We define the discrete l 2 -norm and the discrete maximum norm as   Discrete Dynamics in Nature and Society respectively. Tables 5 and 6 list the errors and convergence rates of the numerical results at t � 10 7 h 4 . From the results, we observe that both the schemes achieve the expected convergence rates (the first-order for LSS (10) and the second-order for CN scheme (11)).

ree-Dimensional Space.
Finally, the two numerical schemes are extended to the three-dimensional CH equation as follows:  Discrete Dynamics in Nature and Society An initial condition is given by We extend the one-dimensional reference solution to the three-dimensional reference numerical solution:     e discrete l 2 -norm and the discrete maximum norm are simply extended as follows: respectively. Tables 7 and 8 list errors and convergence rates for the numerical results at t � 10 6 h 4 . Tables 7 and 8 show that equations (15) and (16) are of first-order and of secondorder in time, respectively.

Conclusions
In this work, we proposed a very simple benchmark problem, which is a cosine function as the initial condition for the numerical methods of the CH equations. For the numerical solution, we used RK4 for the temporal integration. Using the proposed benchmark problem solution, we performed the convergence tests for LSS and CN schemes and obtained the expected first-order and second-order convergence rates in the one-, two-, and three-dimensional CH equations, respectively. e proposed approach is less biased and provides more rational results compared to conventional self-convergence tests. In future work, we will      design simple benchmark problems for other nonlinear partial differential equations such as the nonlocal CH, AC, and nonlocal AC equations.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.