An operator splitting scheme for numerical simulation of spinodal decomposition and microstructure evolution of binary alloys

Abstract

This article compares the operator splitting scheme to linearly stabilized splitting and semi-implicit Euler's schemes for the numerical solution of the Cahn-Hilliard equation. For the purpose of validation, the spinodal decomposition phenomena have been simulated. The efficacy of the three schemes has been demonstrated through numerical experiments. The computed results show that the schemes are conditionally stable. It has been observed that the operator splitting scheme is computationally more efficient.

1. Introduction

The Cahn-Hilliard (CH) equation is a nonlinear parabolic partial differential equation that describes phase transitions in binary alloys, such as spinodal decomposition [1], [2]. The spinodal decomposition is a process for separating a fluid into multiple phases with different physical and chemical properties.

In order to fully understand the slow and rapid dynamics of the spinodal decomposition phenomena, a robust numerical algorithm is required to track the evolution of the moving interface. Several numerical techniques such as Phase-field methods [3], [4], [5], [6], [7], [8], Immersed interface methods [9], [10], Level-set methods [11], [12], Volume-of-fluid methods [13], [14], Immersed boundary method [15], [16], Finite element methods [17], [18], [19], Finite difference method [20], and Boundary integral methods [21], [22], [23] have been developed for interface dynamics in two-phase flow.

The two important properties of Cahn-Hilliard equation are mass conservation and total energy dissipation. Many researchers derived different numerical schemes of various order that satisfies discrete energy and mass conservation at any time. Eyre [24] developed convex splitting scheme for both Cahn-Hilliard and Allen-Cahn equations which is unconditionally stable but its accuracy is of first order. Later Chen et al. [25] developed second order while Cheng et al. [26] constructed fourth order finite difference method for the Cahn-Hilliard equation. Some high-order and stable methods are proposed in the recent work in literature [27], [28], [29], [30]. The second-order accurate and energy stable schemes using pseudo-spectral and operator splitting scheme were proposed for the solution of Cahn-Hilliard equations [31], [32], [33].

In this paper, we mainly focus on three schemes such as the linearly stabilized splitting (LSS), the operator splitting (OPSP) and the semi-implicit Euler's (SIE) schemes [34]. The random initial condition has been taken to show the stability, accuracy and efficiency of these schemes. The numerical investigation shows that the three schemes are conditionally stable. Also, the OPSP and the SIE schemes are more accurate than the LSS scheme. As far as efficiency is concerned, the OPSP scheme is the most efficient.

The remainder of the paper is laid out as follows: Governing equation is given in Section 2. Numerical discretization and stability of the schemes is discussed in section 3. Section 4 contains the computed results. The conclusion is given in Section 5.

2. Governing equations

In this paper, we consider the CH equation given by [35],

$${\partial }_{t}w(\mathbf{x},t)=M\mathrm{\Delta }\mu ,\mathbf{x}\in \mathrm{\Omega },t>0,\mu =-{ϵ}^2\mathrm{\Delta }w(\mathbf{x},t)+g\left(w\right),$$ (1)

with the following initial and boundary conditions:

$$w(\mathbf{x},t)=w_0\left(\mathbf{x}\right),\mathrm{\nabla }w\cdot \hat{n}=0,\mathrm{\nabla }\mu \cdot \hat{n}=0.$$

In Eq. (1), Ω represents the concentration domain, M is the mobility, ϵ is related to the energy of the interface in two-phase flow called the coefficient of the gradient energy, $w(\mathbf{\text{x}},t)$ is the order variable which represent the concentration, and $\mu =g\left(w\right)-{ϵ}^2\mathrm{\Delta }w$ is the chemical potential energy in which $g(w)=w^3-w={\mathcal{F}}^{\prime }(w)$ where the double-well potential function $\mathcal{F}\left(w\right)$ is given by

$$\mathcal{F}\left(w\right)=\frac{1}{4}{(w^2-1)}^2.$$

The CH equation is also known as the phase-field equation which is derived by minimizing the total free energy functional

$$\mathcal{E}(w)=\underset{\mathrm{\Omega }}{\int }[\mathcal{F}(w)+\frac{{ϵ}^2}{2}{\left|\mathrm{\nabla }w\right|}^2]d\mathbf{\text{x}}.$$ (2)

It is well known that the solution of the CH equation $w(\mathbf{x},t)$ has the property that the total free energy $\mathcal{E}(w)$ decreases with time [24] i.e.,

$$\frac{d\mathcal{E}(w)}{dt}\le 0.$$ (3)

The energy functional given by Eq. (2), the Cahn-Hilliard equation (1) with the given boundary conditions satisfies the following energy dissipation property

$$\frac{d\mathcal{E}}{dt}=-M{\Vert \mathrm{\nabla }\mu \Vert }^2.$$

Taking the time derivative of the energy functional (2), then integrating by parts and using the boundary conditions, we have

$$\frac{d\mathcal{E}}{dt}=\underset{\mathrm{\Omega }}{\int }{ϵ}^2\mathrm{\nabla }w\cdot {\partial }_t\left(\mathrm{\nabla }w\right)+{\mathcal{F}}^{\prime }\left(w\right){\partial }_{t}wd\mathbf{\text{x}}=\underset{\mathrm{\Omega }}{\int }{ϵ}^2\mathrm{\nabla }w\cdot \mathrm{\nabla }\left({\partial }_{t}w\right)+{\mathcal{F}}^{\prime }\left(w\right){\partial }_{t}wd\mathbf{\text{x}}={ϵ}^2\underset{\partial \mathrm{\Omega }}{\int }(\mathrm{\nabla }w\cdot \hat{n}){\partial }_{t}wds-\underset{\mathrm{\Omega }}{\int }{ϵ}^2{\mathrm{\nabla }}^{2}w{\partial }_{t}w+{\mathcal{F}}^{\prime }\left(w\right){\partial }_{t}wd\mathbf{\text{x}}=\underset{\mathrm{\Omega }}{\int }(-{ϵ}^2{\mathrm{\nabla }}^{2}w+{\mathcal{F}}^{\prime }(w)){\partial }_{t}wd\mathbf{\text{x}}=M\underset{\mathrm{\Omega }}{\int }\mu {\mathrm{\nabla }}^2\mu d\mathbf{\text{x}}=M\underset{\partial \mathrm{\Omega }}{\int }(\mathrm{\nabla }\mu \cdot \hat{n})\mu ds-M\underset{\mathrm{\Omega }}{\int }\mathrm{\nabla }\mu \cdot \mathrm{\nabla }\mu d\mathbf{\text{x}}=-M{\Vert \mathrm{\nabla }\mu \Vert }^2.$$

Also, $w(\mathbf{x},t)$ is a conserved variable [36], [37] such that

$$\frac{d}{dt}\underset{\mathrm{\Omega }}{\int }w(\mathbf{x},t)d\mathbf{\text{x}}=0.$$ (4)

By taking $M=1$, Eq. (1) becomes

$${\partial }_{t}w(\mathbf{x},t)=\mathrm{\Delta }(g\left(w\right)-{ϵ}^2\mathrm{\Delta }w(\mathbf{x},t)).$$ (5)

Eq. (5) is a fourth-order nonlinear and stiff partial differential equation that is difficult to solve analytically [38].

3. Numerical discretization

In this section, the numerical schemes for discretization of the CH equation in both space and time are given.

3.1. Spatial discretization

The Fourier-spectral approximation has been used for the spatial discretization of Eq. (1). In all directions, uniform Cartesian meshes are used. The domain $\mathrm{\Omega }=[a,b]\times [c,d]$ is divided into $M\times N$ square cells with $M=\frac{b-a}{\mathrm{\Delta }x},N=\frac{d-c}{\mathrm{\Delta }y}$.

Let ${\mathrm{\Omega }}_h=(x_i,y_j):x_i=(i-0.5)\mathrm{\Delta }x,y_j=(j-0.5)\mathrm{\Delta }y,1\le i\le M,1\le j\le N$, be the set of cell-centered points. The approximations for $w(x_i,y_j,n\mathrm{\Delta }t)$ is denoted by $w_{i,j}^n$ where T is the final time, n is the total number of time steps and $\mathrm{\Delta }t=\frac{T}{n}$ is the time step. The main theme of the Fourier-spectral approximation is to transform the given partial differential equation into a system of ordinary differential equations in the Fourier space. The discrete cosine transform of a function $w(x_i,y_j,t_n)$ is defined as:

$${\hat{w}}_{r,s}^n={\alpha }_r{\beta }_s\sum _{i=1}^M\sum _{j=1}^N{w}_{i,j}^n\cos \left(\frac{(2i-1)(r-1)\pi }{2M}\right)\cos \left(\frac{(2j-1)(s-1)\pi }{2N}\right),r=1,2,\cdots ,M,s=1,2,\cdots ,N,$$ (6)

where the scaling parameters are

$${\alpha }_r=\{\begin{array}{cc}\sqrt{\frac{1}{M}}\hfill & r=1,\hfill \\ \sqrt{\frac{2}{M}}\hfill & 2\le r\le M,\hfill \end{array}\text{ and }{\beta }_s=\{\begin{array}{cc}\sqrt{\frac{1}{N}}\hfill & s=1,\hfill \\ \sqrt{\frac{2}{N}}\hfill & 2\le s\le N.\hfill \end{array}$$

For simplicity, we define the following parameters

$$x_i=\frac{(2i-1)b}{2M},y_j=\frac{(2j-1)d}{2N},{\xi }_r=\frac{(r-1)}{b},{\eta }_s=\frac{(s-1)}{d},$$

so that Eq. (6) becomes

$${\hat{w}}_{r,s}^n={\alpha }_r{\beta }_s\sum _{i=1}^M\sum _{j=1}^N{w}_{i,j}^n\cos \left(x_i\pi {\xi }_r\right)\cos \left(y_j\pi {\eta }_s\right),$$

and its inverse discrete transform is

$$w_{i,j}^n=\sum _{r=1}^M\sum _{s=1}^N{\alpha }_r{\beta }_s{\hat{w}}_{r,s}^n\cos \left({\xi }_r\pi x_i\right)\cos \left({\eta }_s\pi y_j\right).$$ (7)

Let

$$w(x,y,n\mathrm{\Delta }t)=\sum _{r=1}^M\sum _{s=1}^N{\alpha }_r{\beta }_s{\hat{w}}_{r,s}^n\cos \left({\xi }_r\pi x\right)\cos \left({\eta }_s\pi y\right),$$

then, the second-order partial derivatives with respect to x and y are given by

$$w_{xx}=-\sum _{r=1}^M\sum _{s=1}^N{\left({\xi }_r\pi \right)}^2{\alpha }_r{\beta }_s{\hat{w}}_{r,s}^n\cos \left({\xi }_r\pi x\right)\cos \left({\eta }_s\pi y\right),$$

$$w_{yy}=-\sum _{r=1}^M\sum _{s=1}^N{\left({\eta }_s\pi \right)}^2{\alpha }_r{\beta }_s{\hat{w}}_{r,s}^n\cos \left({\xi }_r\pi x\right)\cos \left({\eta }_s\pi y\right).$$

Therefore

$$\mathrm{\Delta }w=w_{xx}+w_{yy}=-\sum _{r=1}^M\sum _{s=1}^N({({\xi }_r\pi )}^2+{({\eta }_s\pi )}^2){\alpha }_r{\beta }_s{\hat{w}}_{r,s}^n\cos \left({\xi }_r\pi x\right)\cos \left({\eta }_s\pi y\right).$$

Eq. (2) in discrete form is written as follows:

$${\mathcal{E}}^h\left(w^n\right)=\mathrm{\Delta }{x}^2\sum _{i=1}^M\sum _{j=1}^N\mathcal{F}\left(w_{i,j}^n\right)+\frac{{ϵ}^2}{2}(\sum _{i=0}^M\sum _{j=1}^N{(w_{i+1,j}^n-w_{i,j}^n)}^2+\sum _{i=1}^M\sum _{j=0}^N{(w_{i,j+1}^n-w_{i,j}^n)}^2).$$

3.2. Time discretization

In this subsection, the following three schemes are discussed for solving Eq. (5).

3.2.1. Linearly Stabilized Splitting (LSS) scheme

Applying the LSS scheme [39], [40] to Eq. (5) gives

$$\frac{w_{i,j}^{n+1}-w_{i,j}^n}{\mathrm{\Delta }t}=\mathrm{\Delta }(g(w_{i,j}^n)-2w_{i,j}^n+2w_{i,j}^{n+1}-{ϵ}^2\mathrm{\Delta }{w}_{i,j}^{n+1}).$$ (8)

Applying the discrete cosine transform to Eq. (8) and rearranging, we get

$${\hat{w}}_{r,s}^{n+1}=\frac{{\hat{w}}_{r,s}^n+A_{r,s}\mathrm{\Delta }t{\hat{g}}_{r,s}^n}{1-2A_{r,s}\mathrm{\Delta }t+{ϵ}^2{A}_{r,s}^2\mathrm{\Delta }t},$$

where

$$A_{r,s}=-({\left({\xi }_r\pi \right)}^2+{\left({\eta }_s\pi \right)}^2).$$

The corresponding function $w_{i,j}^{n+1}$ can be obtained using Eq. (7).

3.2.2. Semi-Implicit Euler's (SIE) scheme

Applying the semi-implicit Euler's scheme [41] to Eq. (5), we get

$$\frac{w_{i,j}^{n+1}-w_{i,j}^n}{\mathrm{\Delta }t}=\mathrm{\Delta }(g(w_{i,j}^n)-{ϵ}^2\mathrm{\Delta }{w}_{i,j}^{n+1}).$$ (9)

Since, $g(w)=w^3-w$, applying the discrete cosine transform to Eq. (9) and rearranging, we get

$${\hat{w}}_{r,s}^{n+1}=\frac{{\hat{w}}_{r,s}^n+A_{r,s}\mathrm{\Delta }t{\hat{g}}_{r,s}^n}{1+{ϵ}^2{A}_{r,s}^2\mathrm{\Delta }t}.$$

The corresponding function $w_{i,j}^{n+1}$ can be obtained using Eq. (7).

As far as the stability of the above two schemes is concerned, they are known to be conditionally gradient stable i.e., the numerical solution obtained satisfies Eq. (3) for the CH equation for the particular value of Δt. Moreover, the details regarding their stability are discussed in [24], [42].

3.2.3. Operator Splitting (OPSP) scheme

Consider a general evolution equation of the form

$${\partial }_{t}w=\mathit{\text{L}}(w)+\mathit{\text{N}}(w),w(x,y,0)=w_0(x,y),$$ (10)

where L and N are linear and nonlinear operators respectively. For the CH equation, we choose

$$\mathit{\text{L}}(w)=-{ϵ}^2{\mathrm{\Delta }}^{2}w-\mathrm{\Delta }w,\text{and}\mathit{\text{N}}(w)=\mathrm{\Delta }{w}^3.$$

The main idea of the OPSP scheme is to solve Eq. (10) in two steps. First the linear part is solved analytically and second, the analytical solution is used as an initial condition for solving the nonlinear part. By rewriting Eq. (5)

$${\partial }_{t}w=\mathrm{\Delta }(w^3-w-{ϵ}^2\mathrm{\Delta }w),$$

and splitting it into linear and nonlinear equations [43];

$${\partial }_{t}w=-{ϵ}^2{\mathrm{\Delta }}^{2}w-\mathrm{\Delta }w,$$ (11a)

$${\partial }_{t}w=\mathrm{\Delta }{w}^3.$$ (11b)

These equations are solved in the given order. Taking cosine transform of Eq. (11a), we get:

$$\frac{\partial {\hat{w}}_{r,s}}{\partial t}=-{ϵ}^2{A}_{r,s}^2{\hat{w}}_{r,s}+A_{r,s}{\hat{w}}_{r,s},$$

which has an intermediate solution in the Fourier space as follows:

$${\hat{v}}_{r,s}^n=\exp (-{ϵ}^2{A}_{r,s}^2+A_{r,s})\mathrm{\Delta }t{\hat{w}}_{r,s}^n.$$ (12)

Taking inverse transform of Eq. (12), we get the solution in the physical space as,

$$v_{i,j}^n=C^{-1}[\exp (-{ϵ}^2{A}_{r,s}^2+A_{r,s})\mathrm{\Delta }tC\left[w_{i,j}^n\right]].$$ (13)

Taking the cosine transform of Eq. (11b), we obtain

$$\frac{\partial {\hat{w}}_{r,s}}{\partial t}=-A_{r,s}{\widehat{w^3}}_{r,s}$$

whose solution in the physical space is obtained after taking the inverse cosine transform and is given by

$$\frac{\partial w_{i,j}}{\partial t}=C^{-1}[-A_{r,s}C{\left(w_{i,j}^n\right)}^3].$$ (14)

Applying the forward Euler method for time discretization, we have

$$\frac{w_{i,j}^{n+1}-w_{i,j}^n}{\mathrm{\Delta }t}=C^{-1}[-A_{r,s}C{\left(w_{i,j}^n\right)}^3].$$

Using the intermediate solution Eq. (13) as initial condition i.e. $w_{i,j}^n\approx v_{i,j}^n$, we obtain the following equation

$$w_{i,j}^{n+1}=v_{i,j}^n+\mathrm{\Delta }t\left(C^{-1}[-A_{r,s}C{\left(v_{i,j}^n\right)}^3]\right).$$

In order to solve a time dependent problem, it is first discretized with respect to space, resulting in a system of ordinary differential equations. Then, the system is solved by a finite difference method in time. If we want to investigate the eigenvalue stability of this process, we should find the conditions under which the eigenvalues of the spectral differentiation operator are contained in the stability region of the time discretization formula.

In the operator splitting scheme the linear time dependent problem Eq. (11a) is solved in Fourier space resulting in Eq. (12) which gives us the amplification factor:

$${\mathrm{\Theta }}_{r,s}=\exp (-{ϵ}^2{A}_{r,s}^2+A_{r,s})\mathrm{\Delta }t.$$

According to Von Neumann condition, the method is stable if the amplification factor satisfies $|{\mathrm{\Theta }}_{r,s}|\le 1$. This indicates

$$|{\mathrm{\Theta }}_{r,s}|=|\exp (-{ϵ}^2{A}_{r,s}^2+A_{r,s})\mathrm{\Delta }t|\le 1,$$

$$(-{ϵ}^2{A}_{r,s}^2+A_{r,s})\mathrm{\Delta }t\le 0.$$

$${ϵ}^2{A}_{r,s}^2-A_{r,s}\ge 0.$$ (15)

The last inequality gives us a restriction on the parameters ϵ and $A_{r,s}$, and not on Δt. Now, for the stability of the nonlinear equation i.e., Eq. (11b), we find the eigenvalues of the spectral differentiation matrix.

Theorem[44]: Suppose the grid points N is even and m denotes the order of spatial differentiation. If m is odd the m-th order spatial differentiation matrix$D^{(m)}$is a skew-symmetric matrix with eigenvalues$[-i{(\frac{\mathit{\pi }}{\mathrm{\Delta }x}-1)}^m,i{(\frac{\mathit{\pi }}{\mathrm{\Delta }x}-1)}^m]$, and norm$\Vert D^{(m)}\Vert ={(\frac{\mathit{\pi }}{\mathrm{\Delta }x}-1)}^m$. If m is even then$D^{(m)}$is a symmetric matrix with eigenvalues${(-1)}^{\frac{m}{2}}\times [0,{\left(\frac{\mathit{\pi }}{\mathrm{\Delta }x}\right)}^m]$and norm$\Vert D^{(m)}\Vert ={\left(\frac{\mathit{\pi }}{\mathrm{\Delta }x}\right)}^m$.

Applying above theorem to Eq. (11b) whose spectral differentiation matrix is $D_N^{(2)}$, the eigenvalues are all real given by $[-{(\frac{\pi }{\mathrm{\Delta }x})}^2,0]$. According to the stability restriction all these eigenvalues should lie in the circle. In our case, it should be the real axis within the unit circle [44] (since we are using the forward Euler method for solving Eq. (14). Hence, the stability condition is obtained ${\left(\frac{\pi }{\mathrm{\Delta }x}\right)}^2\le \frac{2}{\mathrm{\Delta }t}$, which gives $\mathrm{\Delta }t\le {\left(\frac{\sqrt{2}\mathrm{\Delta }x}{\pi }\right)}^2$. Hence, in order to keep our method stable, the following conditions should both be guaranteed.

$${ϵ}^2{A}_{r,s}^2-A_{r,s}\ge 0\mathtt{and}\mathrm{\Delta }t\le {\left(\frac{\sqrt{2}\mathrm{\Delta }x}{\pi }\right)}^2.$$ (16)

4. Results and discussion

We perform different numerical experiments in this section to establish numerical stability, search for optimal time-step and compare the efficiency of the three schemes.

4.1. Numerical stability and optimal time-step

The stability of a numerical scheme for CH equation requires that the discrete problem satisfies the energy dissipation property given by Eq. (3), i.e., for $\mathrm{\Delta }t>0$ and for any initial data ${\mathcal{E}}^h(w^{n+1})\le {\mathcal{E}}^h(w^n)$. For validation, consider the initial conditions $w_1(x,y,0)=rand(x,y)$, and $w_2(x,y,0)=6.0\times w_1(x,y,0)$ in the unit square domain with uniform grids of size $h=\frac{1}{64}$. Let $\mathrm{\Delta }{t}_{opt}$ be the optimal time-step for which the above three schemes satisfy the discrete energy dissipation property. From numerical investigation shown in Table 1, it is observed that all the schemes are unstable when $\mathrm{\Delta }t>\mathrm{\Delta }{t}_{opt}$. Further, the LSS scheme is unconditionally stable for $w_1$ and it is conditionally stable for $w_2$. The stability of the OPSP scheme is unaffected by the initial condition and the value of $\mathrm{\Delta }{t}_{opt}$ satisfies the condition (16). However, the SIE scheme has different optimal time-steps for different initial conditions as given in Table 1.

Table 1.

Δt_opt for the three schemes using different initial conditions.

IC	LSS	OPSP	SIE
w1	∞	5 × 10−5	2 × 10−4
w2	2 × 10−4	5 × 10−5	1 × 10−5

4.2. Accuracy

In this section, we discuss the accuracy of the three schemes. For this purpose, we consider the following initial condition $w_3(x,y,0)=0.001\times (w_1(x,y,0)-0.5)$ on a square domain $\mathrm{\Omega }=[0,1]\times [0,1]$. We take $ϵ=0.01$, uniform grids with size $\frac{1}{128}$ and the energy obtained with $\mathrm{\Delta }t={10}^{-10}$ as the reference energy ($E^{ref}$) for each method. The normalized discrete total energy $E(t)/E(0)$ using the three schemes with different time steps is shown in Fig. 1(a-c). It can be seen that at the final time T using different time steps, the energies obtained by the SIE scheme Fig. 1(b) and the OPSP scheme Fig. 1(c) are almost the same as $E^{ref}$, whereas by using the LSS scheme Fig. 1(a) with large time steps significant differences can be seen from the reference energy. This indicates that the LSS scheme is not consistent with long time-steps and did not result in considerable system evolution.

Figure 1.

Energy evolution over time with different time steps using (a) LSS, (b) SIE and (c) OPSP schemes.

From Table 1 and Figs. 1(a-c), it can be concluded that the LSS scheme performs better in term of stability while the OPSP scheme is the most accurate.

4.3. Numerical simulations

We present simulations of spinodal decomposition to assess the efficiency of the schemes, and the results are compared at the nearest energy level (with various time steps) rather than at the same time. In this case, the initial condition $w_3$ is used to solve the CH equation on computational domain $\mathrm{\Omega }=[0,128]\times [0,128]$ with $ϵ=0.01$ and $h=\frac{1}{128}$ using all the three schemes. The results are recorded in Table 2, Table 3, Table 4 respectively. These tables show the time taken by each scheme to reach at the same energy levels represented by $E_1$, $E_2$, $E_3$ and $E_4$ with different time steps. We consider four different time steps $\mathrm{\Delta }{t}_1=1\times {10}^{-6}$, $\mathrm{\Delta }{t}_2=9\times {10}^{-6}$, $\mathrm{\Delta }{t}_3=1\times {10}^{-5}$ and $\mathrm{\Delta }{t}_4=4\times {10}^{-5}$ respectively. From these tables, it can be observed that in order to get to the closest energy levels (with different time steps), the required numerical times are different for the LSS scheme, whereas for the OPSP scheme the numerical times are much better matched but significant differences can be seen for the SIE scheme. Figure 2, Figure 3, Figure 4 show the evolution of $w(\mathbf{x},t)$ with $\mathrm{\Delta }{t}_4$ using LSS, OPSP and SIE schemes respectively.

Table 2.

Time variation for the LSS scheme in order to get to the energy levels E₁, E₂, E₃, and E₄.

Time steps	$E_1=0.958$	$E_2=0.792$	$E_3=0.689$	$E_4=0.619$
	t1 × 10−3	t2 × 10−3	t3 × 10−3	t4 × 10−3
Δt1 = 1 × 10−6	2.5	3	3.25	3.5
Δt2 = 9 × 10−6	2.736	3.276	3.528	3.744
Δt3 = 1 × 10−5	2.79	3.34	3.61	3.85
Δt4 = 4 × 10−5	3.68	4.4	4.72	4.92

Table 3.

Time variation for the OPSP scheme in order to get to the energy levels E₁, E₂, E₃, and E₄.

Time steps	$E_1=0.958$	$E_2=0.792$	$E_3=0.689$	$E_4=0.619$
	t1 × 10−3	t2 × 10−3	t3 × 10−3	t4 × 10−3
Δt1 = 1 × 10−6	2.5	3	3.25	3.5
Δt2 = 9 × 10−6	2.5	3.015	3.267	3.546
Δt3 = 1 × 10−5	2.5	3.01	3.26	3.52
Δt4 = 4 × 10−5	2.48	3	3.28	3.64

Table 4.

Time variation for the SIE scheme in order to get to the energy levels E₁, E₂, E₃, and E₄.

Time steps	$E_1=0.958$	$E_2=0.792$	$E_3=0.689$	$E_4=0.619$
	t1 × 10−3	t2 × 10−3	t3 × 10−3	t4 × 10−3
Δt1 = 1 × 10−6	2.5	3	3.25	3.5
Δt2 = 9 × 10−6	2.574	3.078	3.33	3.582
Δt3 = 1 × 10−5	2.57	3.07	3.31	3.55
Δt4 = 4 × 10−5	2.8	3.36	3.6	3.84

Figure 2.

Evolution of w(x,t) with Δt₄ using LSS scheme.

Figure 3.

Evolution of w(x,t) with Δt₄ using OPSP scheme.

Figure 4.

Evolution of w(x,t) with Δt₄ using SIE scheme.

To compare the evaluation of the energy dissipation for the three schemes, we consider the initial condition $w_4(x,y,0)=0.03\cos (2\pi x)\cos (2\pi y)+0.01\cos (\pi x)\cos (3\pi y)$ [36]. The time-step is taken as $\mathrm{\Delta }t=0.1\times {10}^{-4}$.

The Fig. 5 shows that energy decays faster with the OPSP scheme than with the other two schemes. This shows that the OPSP scheme is more efficient than the LSS and SIE schemes.

Figure 5.

Evolution of energy using the three schemes.

4.4. Conservation of mass

As stated earlier, the CH equation satisfies the mass conservation property, i.e., Eq. (4). In order to show this behavior numerically, we consider the evolution of two adjacent droplets in $\mathrm{\Omega }=(0,2)\times (0,2)$. In this case, the initial conditions are defined as

$$w_5(x,y,0)=\tanh \left(\frac{0.28-\sqrt{{(x-0.7)}^2+{(y-1)}^2}}{2\sqrt{2}{ϵ}^2}\right)+\tanh \left(\frac{0.28-\sqrt{{(x-1.3)}^2+{(y-1)}^2}}{2\sqrt{2}{ϵ}^2}\right).$$

In Figs. 6, we have plotted the mass for different times for all the three schemes. The morphology evolution at different times for the schemes is also represented in these figures. The simulations are carried out for the final time $T=0.3$ with time step $\mathrm{\Delta }t=1\times {10}^{-5}$ and $ϵ=0.01$. These figures show that the mass remains conserved in each case.

Figure 6.

Evolution of two adjacent droplets at different times for (a) LSS (b) OPSP and (c) SIE. The straight line represents that the mass is conserved.

5. Conclusion

In this work, we have solved the nonlinear CH equation numerically using the LSS, SIE, and OPSP schemes. In the OPSP scheme, the CH equation is split into linear and nonlinear sub-equations. The linear part is solved analytically and is used as an initial condition for the numerical solution of the nonlinear sub-equation. Finally, the computed solution by the OPSP scheme is compared with the LSS and SIE scheme. It is concluded that all three schemes are energy stable however the operator splitting scheme is more efficient and less sensitive to time step restriction.

CRediT authorship contribution statement

Abdullah Shah: Conceived and designed the experiments; Wrote the paper; Sana Ayub: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Wrote the paper; Muhammad Sohaib: Analyzed and interpreted the data; Wrote the paper; Sadia Saeed: Performed the experiments; Saher Akmal Khan: Analyzed and interpreted the data; Suhail Abbas; Said Karim Shah: Contributed reagents, materials, analysis tools or data.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.