# Linear Stability Analysis of the Cahn–Hilliard Equation in Spinodal Region

Authors: Seokjun Ham; Darae Jeong; Hyundong Kim; Chaeyoung Lee; Soobin Kwak; Youngjin Hwang; Junseok Kim
Journal: Journal of Function Spaces
 (2022)

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## Abstract

We study a linear stability analysis for the Cahn–Hilliard (CH) equation at critical and off-critical compositions. The CH equation is solved by the linearly stabilized splitting scheme and the Fourier-spectral method. We define the analytic and numerical growth rates and compare these growth rates with respect to the different average levels. In this study, the linear stability analysis is conducted by classifying three average levels such as zero average, spinodal average, and near critical point levels of free energy function, in the one-dimensional (1D) space. The numerical results provide insight for the dynamics of CH equation at critical and off-critical compositions.


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## Body

## 1. Introduction

The Cahn–Hilliard (CH) equation describes the temporal evolution of the conserved phase-field by the following partial differential equation [1, 2]:
(1)∂ϕx,t∂t=ΔF′ϕx,t−ε2Δϕx,t,x∈Ω,t>0,where ϕx,t is a phase-field at position x and time t, Fϕ=0.25ϕ2−12 is the free energy per unit volume and can be substituted for other free energy functions in numerical experiments, and constant ε determines the thickness of the interface. The CH equation (1) was originally introduced as the mathematical model which describes the spinodal decomposition in a binary mixture. This equation has been applied in diverse fields such as dealloying [3, 4], two-phase fluid flow [5–7], topology optimization [8–10], population dynamics [11], tumor growth [12–14], thin films [15–17], block copolymer [18, 19], and image processing [20, 21]. The CH equation is mass-constrained gradient flow in the dual space H−1 of zero average space of the Ginzburg–Landau free energy,
(2)Eϕ≔∫ΩFϕ+ε22∇ϕ2dx.To confirm the total mass conservation and the energy dissipation, we differentiate the total mass ofϕ and equation (2) with respect to time t and obtain the following results, respectively.
(3)ddt∫Ωϕdx=∫Ωϕtdx=∫ΩΔF′ϕ−ε2Δϕdx=0,(4)ddtEϕ=∫ΩF′ϕϕt+ε2∇ϕ·∇ϕtdx=∫ΩF′ϕ−ε2Δϕϕtdx=−∫Ω∇F′ϕ−ε2Δϕ2dx≤0.The results of equations (3) and (4), respectively, show that the conservation of total mass holds, and the total energy does not increase with respect to time. Figure 1 represents a double-well potential Fϕ, its corresponding first and second derivatives. Here, the spinodal region is defined by the gray-colored region where the second derivative is negative, i.e., F′′ϕ<0. In this region, ϕ is unstable and then phase separation occurs. With these various applications of the CH equation, many researchers have proposed new efficient numerical methods [22–24]. Usually, in the stage of verifying the numerical method, the linear stability analysis is used. In [25], the linear stability was used for local equilibrium of Boltzmann equation. In [26], the stability analysis of the advective CH equation was employed to define the perturbation to investigate the instability of wave packets. In this paper, we shall present a clear standard for this test by defining linear and nonlinear regimes.Figure 1
Schematic illustration of double-well potentialFϕ, F′ϕ, and F′′ϕ.The main purpose of this paper is to numerically investigate the dynamics of phase separation in binary mixtures with different average concentrations using the CH equation in the one-dimensional (1D) space. In addition, we propose criteria for linear regimes with small differences between numerical solutions and linear solutions.This paper is organized as follows. The Fourier-spectral method is introduced in Section2 for numerical solution. Section 3 provides numerical results and simulations with linear stability analysis. Conclusions are discussed in Section 4.


## 2. Numerical Solution

Now, we use the Fourier-spectral method [27] to find the numerical solution for the CH equation (1) in the 1D space Ω=0,L,
(5)∂ϕx,t∂t=ΔF′ϕx,t−ε2Δϕx,t,x∈0,L,t>0.Leth=L/Nx be the grid size in the x-axis directions, where Nx is positive even integer. At the edge points xm=m−1h where m=1,2,⋯,Nx, we simply denote the numerical representation ϕxm,tn into ϕmn. Here, tn=nΔt and Δt is the temporal step size. Also, we set μp=2πp/L for the Fourier-spectral method. Let us consider the discrete Fourier transform ϕ^pn=∑m=1Nxϕmne−iμpxm and the inverse discrete Fourier transform
(6)ϕmn=1Nx∑p=−Nx/2Nx/2−1ϕ^pneiμpxm.For simplicity of notation, we definefϕ=F′ϕ=ϕ3−ϕ. By applying the linearly stabilized splitting scheme [28] into the CH equation (5), we obtain the following discrete CH equation:
(7)ϕmn+1−ϕmnΔt=Δfϕmn−2ϕmn+2ϕmn+1−ε2Δϕmn+1=Δϕmn3−3ϕmn+2ϕmn+1−ε2Δϕmn+1.Lettinggϕ=ϕ3−3ϕ, equation (7) comes into
(8)ϕmn+1−ϕmnΔt=Δgϕmn+2ϕmn+1−ε2Δϕmn+1.Then, we can transform equation (8) into the discrete Fourier space as follows:
(9)ϕ^pn+1−ϕ^pnΔt=μp22ϕ^pn+1+ε2μp2ϕ^pn+1+g^pn.By rearranging the above one, we obtain(10)ϕ^pn+1=ϕ^pn−μp2Δtg^pn1+Δt2μp2+ε2μp4.As a consequence, we can compute the updated numerical solutionϕmn+1 by applying the inverse discrete Fourier transform (6).


## 3. Numerical Experiments

In this section, we define growth rate of the CH equation and compare analytic growth rate with numerical growth rate. We also define an error between the analytic solution and the numerical solution. Through several numerical experiments about the growth rate and the error, we investigate the numerical behaviour at different spinodal points. Then, we propose linear and nonlinear regimes.


### 3.1. Analytic and Numerical Growth Rates

In this section, we shall find the analytic and numerical growth rates. We consider the following three different free energy functions [29], F1, F2, and F3 at ϕ=ϕave.
(11)F1ϕ=0.25ϕ2−12,(12)F2ϕ=ϕ6−ϕ4−ϕ2+1,(13)F3ϕ=0.51+ϕln1+ϕ2+1−ϕln1−ϕ2+1.6951−ϕ2.We linearize each ofF′1, F′2, and F′3 by applying the Taylor expansion for different free energy functions; then, we obtain the following linearized terms. The linearized functions are explained in detail in Appendix.
(14)F′1ϕ≈ϕave3−ϕave+3ϕave2−1ϕ−ϕave,(15)F′2ϕ≈6ϕave5−4ϕave3−2ϕave+215ϕave4−6ϕave2−1ϕ−ϕave,(16)F′3ϕ≈0.5ln1+ϕave1−ϕave−3.39ϕave+121+ϕave+121−ϕave−3.39ϕ−ϕave.Therefore, for the free energy functionsF1, F2, and F3, the corresponding linear Cahn–Hilliard (LCH) equations are as follows:
(17)∂ϕx,t∂t=3ϕave2−1Δϕx,t−ε2Δ2ϕx,t,(18)∂ϕx,t∂t=215ϕave4−6ϕave2−1Δϕx,t−ε2Δ2ϕx,t,(19)∂ϕx,t∂t=121+ϕave+121−ϕave−3.39Δϕx,t−ε2Δ2ϕx,t.To derive the ordinary differential equation for amplitude functionαt, we take ϕx,t=ϕave+αtcosKπx. For the positive even integer K, it can satisfy the periodic boundary condition. Substituting above ϕx,t into LCH equations (17), (18), and (19), we have
(20)α′tcosKπx=1−3ϕave2αtKπ2cosKπx−ε2αtKπ4cosKπx,(21)α′tcosKπx=21+6ϕave2−15ϕave4αtKπ2cosKπx−ε2αtKπ4cosKπx,(22)α′tcosKπx=3.39−121+ϕave−121−ϕaveαtKπ2cosKπx−ε2αtKπ4cosKπx.DividingcosKπx on the both sides of equations (20), (21), and (22), we obtain
(23)α′t=Kπ21−3ϕave2−ε2Kπ2αt,(24)α′t=Kπ22+12ϕave2−30ϕave4−ε2Kπ2αt,(25)α′t=Kπ23.39−121+ϕave−121−ϕave−ε2Kπ2αt.The analytic solutions of ordinary differential equations (23), (24), and (25) are as follows:
(26)αt=α0eλ1t,λ1=Kπ21−3ϕave2−ε2Kπ2,(27)αt=α0eλ2t,λ2=Kπ22+12ϕave2−30ϕave4−ε2Kπ2,(28)αt=α0eλ3t,λ3=Kπ23.39−121+ϕave−121−ϕave−ε2Kπ2.In this paper, we useF1 unless the free energy function is mentioned; then, we write F1=F, λ1=λ. In the numerical approach, numerical growth rate is defined by
(29)λ~=1TlogϕNt−ϕave∞α0.Here,·∞ denotes the maximum norm.


### 3.2. Convergence Test

In this section, we verify convergence of the numerical growth rate. For numerical test, we useϕx,0=ϕave+0.001cos10πx, ϕave=0, and ε=0.02 in Ω=0,1.Table1 shows the numerical growth rate λ~ at time T=0.004 for various Δt=T/Nt and h=1/Nx. Here, the analytic growth rate is λ=597.3241. In Table 1, we can see that the numerical growth rate converges to the analytic one as Δt and h decrease.Table 1
Numerical growth rate for variousΔt and h at T=0.004 (λ=597.3241).
Nt\Nx2324252627211304.2649594.2164594.2139594.2139594.2139212304.5360595.7570595.7545595.7545595.7545213304.6718596.5303596.5278596.5278596.5278214304.7397596.9177596.9152596.9152596.9152215304.7737597.1116597.1091597.1091597.1091216304.7907597.2086597.2061597.2061597.2061217304.7992597.2571597.2546597.2546597.2546


### 3.3. Linear Stability Analysis

From now on, we implement the linear stability analysis with the constant solutionϕ=ϕave. In the every numerical experiment, we use the constant ε=εm=mh/22tanh−10.9 to adjust the interfacial thickness. As long as do not more mentioned, the numerical parameters use such as Nx=128, h=1/Nx, Δt =2e-7, α0=0.001, ε=ε11, and T is total time. In addition, we use three different value ϕave=0,0.4, and 0.6 which are in spinodal region and near critical points of free energy. Note that these values are indicated points A,B, and C in Figure 1. For a positive even integer K≤16, the initial condition is selected as ϕx,0=ϕave+0.001cosKπx in Ω=0,1. First, we investigate analytic and numerical growth rates for different mode numbers K.Before the next tests, we define linear solution as follows:(30)ux,t=ϕave+αtcosKπx,where αt=α0eλt and λ is analytic growth rate, see equation (26). Figure 2 shows the change of the analytic and numerical growth rates with respect to each mode numbers K for ϕave=0, 0.4, and 0.6 in short-time evolution. As we can see, numerical growth rate λ~ estimates analytic growth rate λ well in short time (T=0.004). The growth rates in spinodal region (ϕave=0, ϕave=0.4) have positive values for some mode K. On the other hand (ϕave=0.6), the growth rates are always negative regardless of mode number K and also growth rates are monotonically decreased as mode number K increases. Figures 3(a)–3(c) show the analytic growth rates according to mode numbers K for ϕave=0, 0.4, and 0.6, respectively, where the analytic growth rates λ1, λ2, and λ3 are (26), (27), and (28). However, in the long time evolution, it is difficult to expect the numerical growth rate that fits well with the analytic one. In order to find this reason, we first observe dynamics of the numerical solution ϕx,t and linear solution ux,t. In Figure 4, we can check the behaviour of the numerical and linear solution in short and long time evolution for K=2,4, and 6, when ϕave=0 and ϕave=0.4. Unlike the short-time cases, the long-time numerical solutions in all cases show a lot of difference from the linear solutions. This result means that over time, the CH solution has nonlinearity.Figure 2
Analytic and numerical growth rates of the CH equation with different mode numbers atT=0.004 when (a) ϕave=0 and (b) ϕave=0.4 and ϕave=0.6.
(a)(b)Figure 3
Analytic growth rates of the CH equations for different free energy functionsF1, F2, and F3 at (a) ϕave=0, (b) ϕave=0.4, and (c) ϕave=0.6.
(a)
ϕave=0(b)
ϕave=0.4(c)
ϕave=0.6Figure 4
Numerical solution and theoretical solution at spinodal position (a)ϕave=0 and (b) ϕave=0.4 with different mode.Let us consider Ginzburg–Landau free energy (2) by decomposing it as
(31)Eϕ=E1ϕ+E2ϕ,where
(32)E1ϕ=∫ΩFϕdxandE2ϕ=∫Ωε22∇ϕ2dx.We compare short- and long-time numerical solutionϕx,t when ϕave=0 and ϕave=0.4. In this test, we use mode K=2 and α0=0.1. As shown in Figure 5(a), we can see the nonlinearity in the long-time solution after a certain period of time, regardless of ϕave. Figures 5(b)–5(d) show results of double well potential Fϕ, ε/2∇ϕ2, and Fϕ+ε/2∇ϕ2, respectively. Here, the solid and dashed lines mean the short- and long-time results of each function, respectively. Also, the area of the gray-colored region in Figures 5(b)–5(d) represent the energy E1ϕ, E2ϕ, and the Ginzburg–Landau free energy Eϕ, respectively.Figure 5
Long- and short-time evolutions: (a) dynamics of the CH equation, (b)–(d) terms of Ginzburg–Landau free energyEϕ: Fϕ, ε2/2∇ϕ2, and Fϕ+ε2/2∇ϕ2, respectively.As shown in Figure1, as ϕ moves away from ϕ=0, Fϕ becomes smaller than F0 when −1≤ϕ≤1. For this result, in Figure 5(b), dashed line is located below solid line overall of x. It means that Fϕ becomes smaller over time in the spinodal region. Also, according to fluctuation of ϕ, we can see the property of ε2/2∇ϕ2 in Figure 5(c). However, it can also be confirmed that the value has a slight effect on Fϕ+ε/2∇ϕ2 as shown in Figure 5(d). In conclusion, we know that the CH solution grows with time in the spinodal region, and it has nonlinearity in the existing solution in order to reduce the total free energy.Figures6(a)–6(c) show temporal evolution of E1ϕ, E2ϕ, and Eϕ with different ε. All results of E1ϕ and Eϕ decrease with respect to time t. However, it can be seen that the larger the epsilon, that is, the thicker the interface of numerical solution, the slower the result is expressed.Figure 6
Temporal evolution of (a)E1ϕ, (b) E2ϕ, and (c) Eϕ for each ε: ε9, ε11, and ε15.From now on, we verify that the CH solution has nonlinearity over time. Therefore, when performing a linear stability test, it is accurate to examine short time evolution. Also, based on this fact, we propose the criteria for defining the following linear and nonlinear regimes. To find linear regime where numerical solution estimates linear solution well, we define errorenΔt between numerical and linear solution as follows:
(33)enΔt=1Nx∑m=1Nxuxm,nΔt−ϕmn21/2.We can distinguish linear regime and nonlinear regime based on the user-criteriatol as follows: if enΔt<tol; then, linear regime; otherwise, non-linear regime. We propose a criterion tol using the initial condition ux,0 as follows:
(34)tol=14Nx∑m=1Nxuxm,021/2.Figure7 shows proposed criterion and temporal evolution of error et=nΔt for the time t with dynamic of linear and numerical solutions.Figure 7
Temporal evolution of erroret=nΔt for mode K=2.


## 3.1. Analytic and Numerical Growth Rates

In this section, we shall find the analytic and numerical growth rates. We consider the following three different free energy functions [29], F1, F2, and F3 at ϕ=ϕave.
(11)F1ϕ=0.25ϕ2−12,(12)F2ϕ=ϕ6−ϕ4−ϕ2+1,(13)F3ϕ=0.51+ϕln1+ϕ2+1−ϕln1−ϕ2+1.6951−ϕ2.We linearize each ofF′1, F′2, and F′3 by applying the Taylor expansion for different free energy functions; then, we obtain the following linearized terms. The linearized functions are explained in detail in Appendix.
(14)F′1ϕ≈ϕave3−ϕave+3ϕave2−1ϕ−ϕave,(15)F′2ϕ≈6ϕave5−4ϕave3−2ϕave+215ϕave4−6ϕave2−1ϕ−ϕave,(16)F′3ϕ≈0.5ln1+ϕave1−ϕave−3.39ϕave+121+ϕave+121−ϕave−3.39ϕ−ϕave.Therefore, for the free energy functionsF1, F2, and F3, the corresponding linear Cahn–Hilliard (LCH) equations are as follows:
(17)∂ϕx,t∂t=3ϕave2−1Δϕx,t−ε2Δ2ϕx,t,(18)∂ϕx,t∂t=215ϕave4−6ϕave2−1Δϕx,t−ε2Δ2ϕx,t,(19)∂ϕx,t∂t=121+ϕave+121−ϕave−3.39Δϕx,t−ε2Δ2ϕx,t.To derive the ordinary differential equation for amplitude functionαt, we take ϕx,t=ϕave+αtcosKπx. For the positive even integer K, it can satisfy the periodic boundary condition. Substituting above ϕx,t into LCH equations (17), (18), and (19), we have
(20)α′tcosKπx=1−3ϕave2αtKπ2cosKπx−ε2αtKπ4cosKπx,(21)α′tcosKπx=21+6ϕave2−15ϕave4αtKπ2cosKπx−ε2αtKπ4cosKπx,(22)α′tcosKπx=3.39−121+ϕave−121−ϕaveαtKπ2cosKπx−ε2αtKπ4cosKπx.DividingcosKπx on the both sides of equations (20), (21), and (22), we obtain
(23)α′t=Kπ21−3ϕave2−ε2Kπ2αt,(24)α′t=Kπ22+12ϕave2−30ϕave4−ε2Kπ2αt,(25)α′t=Kπ23.39−121+ϕave−121−ϕave−ε2Kπ2αt.The analytic solutions of ordinary differential equations (23), (24), and (25) are as follows:
(26)αt=α0eλ1t,λ1=Kπ21−3ϕave2−ε2Kπ2,(27)αt=α0eλ2t,λ2=Kπ22+12ϕave2−30ϕave4−ε2Kπ2,(28)αt=α0eλ3t,λ3=Kπ23.39−121+ϕave−121−ϕave−ε2Kπ2.In this paper, we useF1 unless the free energy function is mentioned; then, we write F1=F, λ1=λ. In the numerical approach, numerical growth rate is defined by
(29)λ~=1TlogϕNt−ϕave∞α0.Here,·∞ denotes the maximum norm.


## 3.2. Convergence Test

In this section, we verify convergence of the numerical growth rate. For numerical test, we useϕx,0=ϕave+0.001cos10πx, ϕave=0, and ε=0.02 in Ω=0,1.Table1 shows the numerical growth rate λ~ at time T=0.004 for various Δt=T/Nt and h=1/Nx. Here, the analytic growth rate is λ=597.3241. In Table 1, we can see that the numerical growth rate converges to the analytic one as Δt and h decrease.Table 1
Numerical growth rate for variousΔt and h at T=0.004 (λ=597.3241).
Nt\Nx2324252627211304.2649594.2164594.2139594.2139594.2139212304.5360595.7570595.7545595.7545595.7545213304.6718596.5303596.5278596.5278596.5278214304.7397596.9177596.9152596.9152596.9152215304.7737597.1116597.1091597.1091597.1091216304.7907597.2086597.2061597.2061597.2061217304.7992597.2571597.2546597.2546597.2546


## 3.3. Linear Stability Analysis

From now on, we implement the linear stability analysis with the constant solutionϕ=ϕave. In the every numerical experiment, we use the constant ε=εm=mh/22tanh−10.9 to adjust the interfacial thickness. As long as do not more mentioned, the numerical parameters use such as Nx=128, h=1/Nx, Δt =2e-7, α0=0.001, ε=ε11, and T is total time. In addition, we use three different value ϕave=0,0.4, and 0.6 which are in spinodal region and near critical points of free energy. Note that these values are indicated points A,B, and C in Figure 1. For a positive even integer K≤16, the initial condition is selected as ϕx,0=ϕave+0.001cosKπx in Ω=0,1. First, we investigate analytic and numerical growth rates for different mode numbers K.Before the next tests, we define linear solution as follows:(30)ux,t=ϕave+αtcosKπx,where αt=α0eλt and λ is analytic growth rate, see equation (26). Figure 2 shows the change of the analytic and numerical growth rates with respect to each mode numbers K for ϕave=0, 0.4, and 0.6 in short-time evolution. As we can see, numerical growth rate λ~ estimates analytic growth rate λ well in short time (T=0.004). The growth rates in spinodal region (ϕave=0, ϕave=0.4) have positive values for some mode K. On the other hand (ϕave=0.6), the growth rates are always negative regardless of mode number K and also growth rates are monotonically decreased as mode number K increases. Figures 3(a)–3(c) show the analytic growth rates according to mode numbers K for ϕave=0, 0.4, and 0.6, respectively, where the analytic growth rates λ1, λ2, and λ3 are (26), (27), and (28). However, in the long time evolution, it is difficult to expect the numerical growth rate that fits well with the analytic one. In order to find this reason, we first observe dynamics of the numerical solution ϕx,t and linear solution ux,t. In Figure 4, we can check the behaviour of the numerical and linear solution in short and long time evolution for K=2,4, and 6, when ϕave=0 and ϕave=0.4. Unlike the short-time cases, the long-time numerical solutions in all cases show a lot of difference from the linear solutions. This result means that over time, the CH solution has nonlinearity.Figure 2
Analytic and numerical growth rates of the CH equation with different mode numbers atT=0.004 when (a) ϕave=0 and (b) ϕave=0.4 and ϕave=0.6.
(a)(b)Figure 3
Analytic growth rates of the CH equations for different free energy functionsF1, F2, and F3 at (a) ϕave=0, (b) ϕave=0.4, and (c) ϕave=0.6.
(a)
ϕave=0(b)
ϕave=0.4(c)
ϕave=0.6Figure 4
Numerical solution and theoretical solution at spinodal position (a)ϕave=0 and (b) ϕave=0.4 with different mode.Let us consider Ginzburg–Landau free energy (2) by decomposing it as
(31)Eϕ=E1ϕ+E2ϕ,where
(32)E1ϕ=∫ΩFϕdxandE2ϕ=∫Ωε22∇ϕ2dx.We compare short- and long-time numerical solutionϕx,t when ϕave=0 and ϕave=0.4. In this test, we use mode K=2 and α0=0.1. As shown in Figure 5(a), we can see the nonlinearity in the long-time solution after a certain period of time, regardless of ϕave. Figures 5(b)–5(d) show results of double well potential Fϕ, ε/2∇ϕ2, and Fϕ+ε/2∇ϕ2, respectively. Here, the solid and dashed lines mean the short- and long-time results of each function, respectively. Also, the area of the gray-colored region in Figures 5(b)–5(d) represent the energy E1ϕ, E2ϕ, and the Ginzburg–Landau free energy Eϕ, respectively.Figure 5
Long- and short-time evolutions: (a) dynamics of the CH equation, (b)–(d) terms of Ginzburg–Landau free energyEϕ: Fϕ, ε2/2∇ϕ2, and Fϕ+ε2/2∇ϕ2, respectively.As shown in Figure1, as ϕ moves away from ϕ=0, Fϕ becomes smaller than F0 when −1≤ϕ≤1. For this result, in Figure 5(b), dashed line is located below solid line overall of x. It means that Fϕ becomes smaller over time in the spinodal region. Also, according to fluctuation of ϕ, we can see the property of ε2/2∇ϕ2 in Figure 5(c). However, it can also be confirmed that the value has a slight effect on Fϕ+ε/2∇ϕ2 as shown in Figure 5(d). In conclusion, we know that the CH solution grows with time in the spinodal region, and it has nonlinearity in the existing solution in order to reduce the total free energy.Figures6(a)–6(c) show temporal evolution of E1ϕ, E2ϕ, and Eϕ with different ε. All results of E1ϕ and Eϕ decrease with respect to time t. However, it can be seen that the larger the epsilon, that is, the thicker the interface of numerical solution, the slower the result is expressed.Figure 6
Temporal evolution of (a)E1ϕ, (b) E2ϕ, and (c) Eϕ for each ε: ε9, ε11, and ε15.From now on, we verify that the CH solution has nonlinearity over time. Therefore, when performing a linear stability test, it is accurate to examine short time evolution. Also, based on this fact, we propose the criteria for defining the following linear and nonlinear regimes. To find linear regime where numerical solution estimates linear solution well, we define errorenΔt between numerical and linear solution as follows:
(33)enΔt=1Nx∑m=1Nxuxm,nΔt−ϕmn21/2.We can distinguish linear regime and nonlinear regime based on the user-criteriatol as follows: if enΔt<tol; then, linear regime; otherwise, non-linear regime. We propose a criterion tol using the initial condition ux,0 as follows:
(34)tol=14Nx∑m=1Nxuxm,021/2.Figure7 shows proposed criterion and temporal evolution of error et=nΔt for the time t with dynamic of linear and numerical solutions.Figure 7
Temporal evolution of erroret=nΔt for mode K=2.


## 4. Conclusions

We studied a linear stability analysis for the CH equation in spinodal region as different average levels. To solve the CH equation, we used linearly stabilized splitting scheme and Fourier-spectral method. Through the numerical simulations, we observed various dynamics of the CH equation and confirmed the numerical solution compared linear solution over time. We defined growth rate and also compared numerical growth rate and analytic growth rate. Using difference between numerical and linear solution, we defined error. By means of defined error, we distinguish linear regime where numerical solution and linear solution match because of the small error and nonlinear regime where do not.



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