Stability and Robustness of Time-discretization Schemes for the Allen-Cahn Equation via Bifurcation and Perturbation Analysis

Abstract

The Allen-Cahn equation is a fundamental model for phase transitions, offering critical insights into the dynamics of interface evolution in various physical systems. This paper investigates the stability and robustness of frequently utilized time-discretization numerical schemes for solving the Allen-Cahn equation, with focuses on the Backward Euler, Crank-Nicolson (CN), convex splitting of modified CN, and Diagonally Implicit Runge-Kutta (DIRK) methods. Our stability analysis reveals that the Convex Splitting of the Modified CN scheme exhibits unconditional stability, allowing greater flexibility in time step size selection, while the other schemes are conditionally stable. Additionally, our robustness analysis highlights that the Backward Euler method converges to correct physical solutions regardless of initial conditions. In contrast, all other methods studied in this work show sensitivity to initial conditions and may converge to incorrect physical solutions if the initial conditions are not carefully chosen. This study introduces a comprehensive approach to assessing stability and robustness in numerical methods for solving the Allen-Cahn equation, providing a new perspective for evaluating numerical techniques for general nonlinear differential equations.

1. Introduction

The Allen-Cahn equation, a fundamental partial differential equation (PDE) in the field of phase transitions, describes the process of phase separation in multi-component alloy systems. Its significance extends to numerous applications in materials science [21], image processing [3, 29], and areas requiring the modeling of interface dynamics [2, 15]. Due to the equation’s nonlinearity and the presence of diffuse interface in solutions, developing robust and stable numerical schemes is a long-lasting challenge for accurate simulations [9, 13, 25, 35]. As important as spatial discretization, time discretization is crucial since it also directly determines the efficiency and accuracy of the numerical schemes [10, 11, 30, 31, 41]. Below is an incomplete list of frequently utilized one step explicit and implicit time discretization methods for solving the Allen-Cahn equation and other phase field models:

• Explicit Methods

  • Forward Euler Method: This first-order method approximates the time derivative using a simple forward difference. It is conditionally stable and often requires very small time step sizes, especially for stiff problems like the Allen-Cahn equation [13, 25].
  • Explicit Runge-Kutta Methods: Higher-order explicit methods, such as the fourth-order Runge-Kutta, can be used to improve accuracy while still being conditionally stable [40]. These methods are rarely used due to the stringent time step size restrictions imposed by stability considerations.
• Implicit Methods

  • Backward Euler Method: This first-order implicit method is unconditionally stable and well-suited for stiff problems [13, 25]. However, it often requires solving a nonlinear system at each time step.
  • Crank-Nicolson (CN) Method: This second-order implicit method is based on the trapezoidal rule and offers a good balance between accuracy and stability [14, 23, 41]. It is also unconditionally stable but, like the backward Euler method, requires solving a nonlinear system at each step.
  • Diagonally Implicit Runge-Kutta (DIRK) Methods: These methods are a subclass of implicit Runge-Kutta methods where the coefficient matrix is lower triangular with equal diagonal elements [39]. This structure simplifies the implementation by allowing a step-by-step solution of implicit equations, which improves stability and accuracy while maintaining reasonable computational costs.
• Semi-Implicit Methods

  • Semi-Implicit Spectral Deferred Correction (SISDC) Method: This method iteratively corrects the solution using both implicit and explicit updates, improving stability and accuracy [32].
  • Semi-Implicit Euler Method: This approach involves treating the stiff linear terms implicitly while handling the nonlinear terms explicitly, reducing the complexity of solving fully implicit equations. [5, 35, 36]
  • Convex-splitting Method: This method explores careful splitting of the nonconvex term into the difference of two convex terms, and makes them implicit and explicit, respectively. See [1, 12, 16, 17].

The choice of time discretization method for the Allen-Cahn equation and other phase field models requires careful consideration of stability, accuracy, and computational efficiency [37]. Implicit and semi-implicit methods are often favored for their desired stability properties, and are particularly suited for stiff problems. Notably, these methods require solving nonlinear systems.

Fourier or energy methods are often used to analyze stability conditions for linear schemes of linear partial differential equations with constant coefficients. Yet few have become known about the stability of numerical schemes for nonlinear equations. The study in [38] indicates that many numerical schemes, except for the backward Euler method, may experience convergence issues unless the time step size is exceedingly small. Motivated by this work, we introduce in this paper the following concepts of stability and robustness for numerical schemes designed to solve the Allen-Cahn equation.

Definition 1.1. Stability is defined as the uniqueness of ${\varphi }^{n+1}$ for a given ${\varphi }^n$, revealing the upper bound for the time step size of the numerical scheme which is the stability condition;

Definition 1.2. Robustness is defined as the uniqueness of ${\varphi }^n$ for a given ${\varphi }^{n+1}$, indicating the numerical scheme’s accuracy in converging to the physical solution across various initial conditions.

We note that our notion of stability may also be referred to as unique solvability. It differs from energy stability that does not guarantee the uniqueness of the solution, see [35, 37, 38]. Robustness, on the other hand, refers to the accuracy of a numerical scheme in converging to the correct physical solution with different initial conditions. A non-robust scheme may converge to an incorrect physical solution or an alternative steady-state solution. A lack of robustness indicates that, regardless of the time step size, an initial condition that leads the numerical scheme to converge to an incorrect solution can be constructed. In this paper, we focus on assessing the robustness of schemes by examining the uniqueness of ${\varphi }^n$ for a given ${\varphi }^{n+1}$.

While the stability of numerical schemes is a major concern in analyzing them, importance of the robustness of the nonlinear schemes is sometimes overlooked. In fact, the robustness here indicates the sensitivity of the (nonlinear) schemes to the initial guess used to solve them in each time-stepping. Thus a numerical solution computed using a scheme suffering from robustness issues may converge to a wrong solution.

Both stability and robustness can be investigated through bifurcation theory [6, 24, 28, 33] and perturbation analysis [4, 22], powerful mathematical tools that examine the behavior of solution structures. Stability, as defined in this paper, was previously studied using fixed-point analysis [8], convex energy analysis [37], or analysis of roots in cubic equations [38]. Bifurcation analysis allows ones not only to explore the uniqueness of the numerical solutions but also to identify critical points where qualitative changes in the solution structure occur. Perturbation analysis provides insight into how small perturbations affect the structure of trivial solutions. Together, these analyses form a new and rigorous framework to evaluate the performance of different numerical schemes for phase-field equations.

In this paper, we examine both the stability and robustness of several time-discretization numerical schemes that are commonly used for the Allen-Cahn equation. They include the Backward Euler method, Crank-Nicolson method, and Runge-Kutta methods. Our findings reveal both essential stability conditions and sensitivity to initial conditions based on robustness, guiding the development of efficient and reliable computational methods for simulating the Allen-Cahn equation. Through this investigation, we aim to develop a general framework based on stability and robustness in the numerical treatment of phase field models.

The rest of the paper is organized as follows: In Section 2, we present the Allen-Cahn equation considered in this study. Section 3 explores the stability and robustness of the Backward Euler scheme. In Section 4, we extend this analysis to the Crank-Nicolson scheme. Section 5 examines the convex splitting of the modified Crank-Nicolson scheme. In Section 6, we study the Diagonally Implicit Runge-Kutta method. Section 7 explores all the schemes that we investigated in Sections 3–6 with benchmark problems. Finally, we conclude our work in Section 8.

2. Problem Setup

We consider the following time-dependent Allen-Cahn equation on domain $[-\mathrm{1,1}{]}^d=\Omega \subset {\mathbb{R}}^d$:

$$\left\{\begin{array}{l}{\varphi }_t-\Delta \varphi +\frac{1}{{\epsilon }^2}\left({\varphi }^3-\varphi \right)=0,x\in \Omega ,t\in \left[0,T\right],\\ \varphi \left(x,0\right)={\varphi }_0\left(x\right),x\in \Omega ,\\ \frac{\partial \varphi (x,t)}{\partial \mathbf{n}}=0,x\in \partial \Omega ,t\in [0,T]\end{array}\right.$$ (1)

where $\epsilon$ is a small positive constant representing the thickness of the diffuse interface, ${\varphi }_0\left(x\right)$ is given and $\mathbf{n}$ is the unit outward normal vector to $\partial \Omega ,x={\left(x_1,x_2,\cdots ,x_d\right)}^T$. It is well known that the Allen-Cahn equation possesses two stable steady state solutions $\varphi \left(x\right)=\pm 1$, respectively.

Our objective is to investigate numerical methods for solving the Allen-Cahn equation. We consider nonlinear schemes given in the general formula $F\left({\varphi }^n\left(x\right),{\varphi }^{n+1}\left(x\right)\right)=0$, where ${\varphi }^n\left(x\right)$ represents the solution at time step $n$, and ${\varphi }^{n+1}\left(x\right)$ represents the solution at the next time step $n+1$.

3. Backward Euler Scheme

The backward Euler scheme for the Allen-Cahn Eq. (1) reads as:

$$\left\{\begin{array}{l}\frac{{\varphi }^{n+1}-{\varphi }^n}{\Delta t}-\Delta {\varphi }^{n+1}+\frac{1}{{\epsilon }^2}\left({\left({\varphi }^{n+1}\right)}^3-{\varphi }^{n+1}\right)=0,\\ \frac{\partial {\varphi }^n\left(x\right)}{\partial \mathbf{n}}=\frac{\partial {\varphi }^{n+1}\left(x\right)}{\partial \mathbf{n}}=0,x\in \partial \Omega ,\end{array}\right.$$ (2)

where $\Delta t$ is the time step size. We will derive its stability condition through bifurcation analysis and its robustness through perturbation analysis in the following subsection.

3.1. The stability condition via bifurcation analysis

In this section, we begin with the constant solution of Eq. (2), then perform the bifurcation analysis of the constant solution with respect to the parameter $\epsilon$.

First, we examine a constant solution case of Eq. (2), where ${\varphi }^n\equiv r$ and ${\varphi }^{n+1}\equiv c$, satisfying the equation:

$$\frac{c-r}{\Delta t}+\frac{1}{{\epsilon }^2}\left(c^3-c\right)=0.$$

Next, we consider a general perturbed case by introducing ${\varphi }^n\left(x\right)=r+\delta f\left(x\right)$ and ${\varphi }^{n+1}\left(x\right)=c+\delta \psi \left(x\right)$, with $\delta \in \mathbb{R},\left|\delta \right|\ll 1$, and $f$ and $\psi$ being some smooth functions. Substituting these functions into Eq. (2) and retaining the linear term in $\delta$, we obtain

$$\frac{\delta \psi -\delta f}{\Delta t}-\delta \Delta \psi +\frac{1}{{\epsilon }^2}\left(3c^2-1\right)\delta \psi +O\left({\delta }^2\right)=0.$$ (3)

Rearranging this equation, we obtain

$$-\Delta \psi +\left(\frac{1}{\Delta t}+\frac{3c^2-1}{{\epsilon }^2}\right)\psi +O\left(\delta \right)=\frac{f}{\Delta t}.$$ (4)

We can assess the uniqueness of ${\varphi }^{n+1}$ by studying the solution structure of Eq. (4) after dropping the $O\left(\delta \right)$ term. In this case, $f\left(x\right)$ is a given function, and we focus on examining the homogeneous part of Eq. (4), namely,

$$-\Delta \psi +\left(\frac{1}{\Delta t}+\frac{3c^2-1}{{\epsilon }^2}\right)\psi =0,$$ (5)

which is a Helmholtz equation. We arrive at the following proposition.

Proposition 3.1. Bifurcations of $\psi$ in Eq. (5) occur when $1-3c^2>0$ and $\Delta t>\frac{{ϵ}^2}{1-3c^2}\ge {ϵ}^2$. The bifurcation points and corresponding eigenfunctions are as follows:

$${ϵ}^2=\frac{1-3c^2}{\frac{1}{\Delta t}+\sum _{i=1}^d{\pi }^2{k}_i^2},\psi \left(x\right)=\prod _{i=1}^d{A}_i\left(x_i\right),A_i\left(x_i\right)=\left\{\begin{array}{ll}\text{cos}\left(\pi k_i{x}_i\right),& k_i=\mathrm{0,1},2,\cdots ,\\ \text{sin}\left(\pi k_i{x}_i\right),& k_i=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots ,\end{array}\right.$$ (6)

where the function $A_i\left(x_i\right)$ can either be $\text{cos}\left(\pi k_i{x}_i\right)$ or $\text{sin}\left(\pi k_i{x}_i\right)$ depending the values of $k_i$.

Proof. By the method of separation of variables, we let $\psi \left(x\right)={\prod }_{i=1}^d{A}_i\left(x_i\right)$. Substituting this into Eq. (5) yields:

$$\left(\frac{1}{\Delta t}+\frac{3c^2-1}{{ϵ}^2}\right)\prod _{i=1}^d{A}_i\left(x_i\right)-\sum _{i=1}^d\left({\partial }_{x_i}^2{A}_i\left(x_i\right)\prod _{j\ne i}{A}_j\left(x_j\right)\right)=0.$$

For the trivial solution, this equation holds for any constant $ϵ$. For the non-trivial case where $A_i\ne 0$, we divide both sides by ${\prod }_{i=1}^d{A}_i\left(x_i\right)$ to obtain:

$$\sum _{i=1}^d\frac{{\partial }_{x_i}^2{A}_i\left(x_i\right)}{A_i\left(x_i\right)}=\left(\frac{1}{\Delta t}+\frac{3c^2-1}{{ϵ}^2}\right).$$ (7)

Each term $\frac{{\partial }_{x_i}^2{A}_i\left(x_i\right)}{A_i\left(x_i\right)}$ in Eq. (7) is constant because the sum ${\sum }_{i=1}^d\frac{{\partial }_{x_i}^2{A}_i\left(x_i\right)}{A_i\left(x_i\right)}$ remains constant regardless of changes in the variables $x_i$. This reduces our problem to a 1D eigenvalue problem with the Neumann boundary condition:

$${\partial }_{x_i}^2{A}_i\left(x_i\right)=c_i{A}_i\left(x_i\right),{\left.{\partial }_{x_i}{A}_i\left(x_i\right)\right|}_{x_i=-1}={\left.{\partial }_{x_i}{A}_i\left(x_i\right)\right|}_{x_i=1}=0,i=\mathrm{1,2},\cdots ,d,$$ (8)

where ${\sum }_{i=1}^d{c}_i=\left(\frac{1}{\Delta t}+\frac{3c^2-1}{{ϵ}^2}\right)$.

The eigenvalue $c_i$ and corresponding eigenfunction $A_i\left(x_i\right)$ for (8) can be easily computed, and are given by:

$$c_i=k_i^2{\pi }^2,A_i\left(x_i\right)=\left\{\begin{array}{ll}\text{cos}\left(\pi k_i{x}_i\right),& k_i=\mathrm{0,1},2,\cdots ,\\ \text{sin}\left(\pi k_i{x}_i\right),& k_i=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots ,\end{array}\right.$$ (9)

where the eigenfunction $A_i\left(x_i\right)$ can either be $\text{cos}\left(\pi k_i{x}_i\right)$ or $\text{sin}\left(\pi k_i{x}_i\right)$ depending the values of $k_i$.

To satisfy Eq. (5), we also require ${\sum }_{i=1}^d{c}_i=-{\sum }_{i=1}^d{\pi }^2{k}_i^2=\left(\frac{1}{\Delta t}+\frac{3c^2-1}{{ϵ}^2}\right)$. Thus, we have the bifurcation point ${ϵ}^2=\frac{1-3c^2}{\frac{1}{\Delta t}+{\sum }_{i=1}^d{\pi }^2{k}_i^2}$ corresponding to the eigenfunction ${\prod }_{i=1}^d{A}_i\left(x_i\right)$.

From Proposition 3.1, we can see that, if $\Delta t\le {ϵ}^2$, the solution to Eq. (5) will exclusively exhibit the trivial solution $\psi =0$. Consequently, we can conclude that the particular solution for Eq. (4) is unique, and ${\varphi }^{n+1}$ is also unique while satisfying the stability condition $\Delta t\le {ϵ}^2$. Thus, the backward Euler scheme is stable when $\Delta t\le {ϵ}^2$.

Remark 3.1. For the robustness analysis of the backward Euler scheme, we examine the uniqueness of ${\varphi }^n$ given ${\varphi }^{n+1}$. As Eq. (2) is linear with respect to ${\varphi }^n$, establishing uniqueness is straightforward.

4. Crank-Nicolson Scheme

The Crank-Nicolson scheme for the Allen-Cahn Eq. (1) reads as

$$\left\{\begin{array}{l}\frac{{\varphi }^{n+1}-{\varphi }^n}{\Delta t}-\frac{1}{2}\left(\Delta {\varphi }^{n+1}+\Delta {\varphi }^n\right)+\frac{1}{2{ϵ}^2}\left({\left({\varphi }^{n+1}\right)}^3-{\varphi }^{n+1}\right)+\frac{1}{2{ϵ}^2}\left({\left({\varphi }^n\right)}^3-{\varphi }^n\right)=0,\\ \frac{\partial {\varphi }^n\left(x\right)}{\partial \mathbf{n}}=\frac{\partial {\varphi }^{n+1}\left(x\right)}{\partial \mathbf{n}}=0,x\in \partial \Omega .\end{array}\right.$$ (10)

4.1. The stability condition via bifurcation analysis

In this section, we investigate the uniqueness of ${\varphi }^{n+1}$ for any given ${\varphi }^n$ and derive the associated stability condition. We start by considering a perturbed setup with respect to the trivial solution.

Here we define ${\varphi }^{n+1}=c+\delta \psi \left(x\right)$ and ${\varphi }^n=r+\delta f\left(x\right)$, with $c$ and $r$ representing constant solutions of Eq. (10), and $f\left(x\right)$ being a given function with $\left|\delta \right|\ll 1$. Plugging these expressions into Eq. (10), we obtain:

$$\frac{c+\delta \psi -r-\delta f}{\Delta t}-\frac{1}{2}(\delta \Delta \psi +\delta \Delta f)+\frac{1}{2{ϵ}^2}\left((c+\delta \psi {)}^3-c-\delta \psi \right)+\frac{1}{2{ϵ}^2}\left((r+\delta f{)}^3-r-\delta f\right)=0.$$ (11)

This equation is simplified as

$$-\frac{1}{2}\Delta \psi +\left(\frac{1}{\Delta t}+\frac{3c^2-1}{2{ϵ}^2}\right)\psi +O\left(\delta \right)=G\left(x\right),$$ (12)

where $G\left(x\right)$ is given by:

$$G\left(x\right)=\frac{f\left(x\right)}{\Delta t}+\frac{1}{2}\Delta f\left(x\right)-\frac{1}{2{ϵ}^2}\left(3r^{2}f\left(x\right)-f\left(x\right)\right).$$ (13)

Considering the homogeneous case of Eq. (12) and dropping the $O\left(\delta \right)$ term, we have:

$$-\frac{1}{2}\Delta \psi +\left(\frac{1}{\Delta t}+\frac{3c^2-1}{2{ϵ}^2}\right)\psi =0.$$ (14)

Then we can deduce the following bifurcation results:

Proposition 4.1. Bifurcations of $\psi$ in Eq. (14) occur when $1-3c^2>0$ and $\Delta t>\frac{2{ϵ}^2}{1-3c^2}\ge 2{ϵ}^2$. The bifurcation points and corresponding eigenfunctions are as follows:

$${ϵ}^2=\frac{1-3c^2}{\frac{2}{\Delta t}+\sum _{i=1}^d{\pi }^2{k}_i^2},\psi \left(x\right)=\prod _{i=1}^d{A}_i\left(x_i\right),A_i\left(x_i\right)=\left\{\begin{array}{ll}\text{cos}\left(\pi k_i{x}_i\right),& k_i=\mathrm{0,1},2,\cdots ,\\ \text{sin}\left(\pi k_i{x}_i\right),& k_i=,\frac{3}{2},\frac{5}{2},\cdots ,\end{array}\right.$$ (15)

where the function $A_i\left(x_i\right)$ can either be $\text{cos}\left(\pi k_i{x}_i\right)$ or $\text{sin}\left(\pi k_i{x}_i\right)$ depending the values of $k_i$.

From Proposition 4.1, we can see that, if $\Delta t\le 2{ϵ}^2$, the solution to Eq. (14) will exclusively exhibit the trivial solution $\psi =0$. Consequently, we can conclude that the particular solution for Eq. (12) is unique, and ${\varphi }^{n+1}$ is also unique while satisfying the stability condition $\Delta t\le 2{ϵ}^2$. The proof proceeds by employing similar computations as those used in Proposition 3.1.

4.2. The robustness analysis

In this section, we delve into the solution space of ${\varphi }^n$ given a ${\varphi }^{n+1}$. Initially, we investigate the trivial solutions; subsequently, we apply perturbation analysis to examine these trivial solutions.

4.2.1. Trivial solution analysis

We consider ${\varphi }^{n+1}\equiv c$ and ${\varphi }^n\equiv r$ and rewrite Eq. (10) as:

$$\frac{c-r}{\Delta t}+\frac{1}{2{ϵ}^2}\left(c^3-c\right)+\frac{1}{2{ϵ}^2}\left(r^3-r\right)=0.$$ (16)

By fixing $c$, if $-27{\left(\frac{-2{ϵ}^{2}c}{\Delta t}-c^3+c\right)}^2+4{\left(\frac{2{ϵ}^2}{\Delta t}+1\right)}^3>0$, we have 3 different real solutions for $r$ by the discriminant of cubic polynomial. If it is equal to 0, then we have multiple real solutions. If it is less than 0, we have complex solutions and a real solution.

First, we let $c=0$ and solve Eq. (16) to get two non-zero roots for $r$ and denote them as $\pm r_1$, where $r_1:=\sqrt{1+\frac{2{ϵ}^2}{\Delta t}}$.

Then we have the following results:

1. $c=0$ if and only if $r=0,\pm r_1$;

2. $c>0$ if and only if $r\in \left(0,r_1\right)\cup \left(-\mathrm{\infty },-r_1\right)$;

3. $c<0$ if and only if $r\in \left(-r_1,0\right)\cup \left(r_1,\mathrm{\infty }\right)$.

Therefore, given a time step size $\Delta t\le 2{ϵ}^2$ satisfying the stability condition, to have $\text{sign}\left(c\right)=\text{sign}\left(r\right)$ hold, we must have $\left|r\right|\le r_1$; to have $\text{sign}\left(c\right)\ne \text{sign}\left(r\right)$ hold, we must have $\left|r\right|>r_1$. The above analysis is discussed in Theorem 3.2 of [38] to show that the Crank-Nicolson method may converge to a wrong steady-state solution. We carry out this further to obtain a more insightful convergence pattern.

Next, we compute roots by solving Eq. (16) with $c=r_1$. Since

$$-27{\left(\frac{-2{ϵ}^2{r}_1}{\Delta t}-r_1^3+r_1\right)}^2+4{\left(\frac{2{ϵ}^2}{\Delta t}+1\right)}^3<0,$$ (17)

we can obtain only one negative real solution and denote it as $-r_2$. Since $c=r_1>0$, we have $-r_2\in \left(-\mathrm{\infty },-r_1\right)$ which implies $r_1<r_2$. Moreover, notice that the value of $-27{\left(\frac{-2{ϵ}^{2}c}{\Delta t}-c^3+c\right)}^2+4{\left(\frac{2{ϵ}^2}{\Delta t}+1\right)}^3$ decreases if $c$ is getting larger than $r_1$. Thus we can find a unique sequence of $r_n$ by repeating this process with $c=-r_{i-1}$. This gives us a pattern of signs of trivial solutions at consecutive time steps. We can conclude that:

1. ${\varphi }^{n+1}>0$ if and only if ${\varphi }^n\in \left(0,r_1\right)\cup \left(-\mathrm{\infty },-r_1\right)$;

2. ${\varphi }^{n+2}>0$ if and only if ${\varphi }^n\in \left(0,r_1\right)\cup \left(-r_2,-r_1\right)\cup \left(r_2,\mathrm{\infty }\right)$;

3. ${\varphi }^{n+1}<0$ if and only if ${\varphi }^n\in \left(-r_1,0\right)\cup \left(r_1,\mathrm{\infty }\right)$;

4. ${\varphi }^{n+2}<0$ if and only if ${\varphi }^n\in \left(-r_1,0\right)\cup \left(r_1,r_2\right)\cup \left(-\mathrm{\infty },-r_2\right)$.

Now we have generated a sequence of $r_i$ by solving Eq. (16) with setting $c=-r_{i-1}$. The numerical values of $r_i$ are presented in Table 1 for different values of $\frac{\Delta t}{2{ϵ}^2}$. The convergence intervals are shown in Fig. 1. If one chooses the initial condition ${\varphi }^n$ from the interval $\left[r_i,r_{i+1}\right]$, the CN scheme converges to $(-1{)}^i\times \left[0,r_1\right]$ after $i$ time-stepping.

Table 1:

Convergence Interval points $r_i$ for the CN Scheme in Eq. (10) with different $\frac{\Delta t}{2{ϵ}^2}$.

$\frac{\Delta t}{2{ϵ}^2}$	$r_1$	$r_2$	$r_3$	$r_4$
0.001	31.639	48.124	60.363	70.53
0.01	10.05	15.256	19.123	22.335
0.1	3.317	4.942	6.152	7.159
0.25	2.236	3.243	3.996	4.625
0.5	1.732	2.421	2.941	3.377

Figure 1:

Visualizing Convergence Intervals of CN Scheme in Eq. (10). If initial conditions are chosen in red regions, CN Scheme eventually converges to 1, while the initial conditions chosen in blue regions lead the CN Scheme to converge to −1. The values of $r_n$ with different $\frac{\Delta t}{2{ϵ}^2}$ are shown in Table 1.

4.2.2. Perturbation Analysis

In this section, we delve into non-trivial solutions by perturbing the trivial solutions analyzed in the previous section. Specifically, we define ${\varphi }^{n+1}=c+\delta f\left(x\right)$ and ${\varphi }^n=r+\delta \psi \left(x\right)$. Our objective is to investigate ${\varphi }^n$ for a given ${\varphi }^{n+1}$. In this case, $f\left(x\right)$ is a given perturbation function, such as $\text{cos}\left(k\pi x\right)$ in the 1D case. We need to solve for $\psi \left(x\right)$. To achieve this, we substitute these functions into Eq. (10), resulting in:

$$-\frac{1}{2}\Delta \psi +\left(-\frac{1}{\Delta t}+\frac{3r^2-1}{2{ϵ}^2}\right)\psi +O\left(\delta \right)=G\left(x\right),$$ (18)

where $G\left(x\right)$ is defined as

$$G\left(x\right)=-\frac{f\left(x\right)}{\Delta t}+\frac{1}{2}\Delta f\left(x\right)-\frac{1}{2{ϵ}^2}\left(3c^{2}f\left(x\right)-f\left(x\right)\right).$$ (19)

To be more specific for the 1D case, when we choose $f\left(x\right)=\text{cos}\left(k\pi x_1\right)$ for $k\in {\mathbb{N}}^{+}$, we find that $\psi =B\text{cos}\left(k\pi x_1\right)$, with the coefficient $B$ given by

$$B=-\frac{\frac{3c^2}{{ϵ}^2}-\frac{1}{{ϵ}^2}+\frac{2}{\Delta t}+(k\pi {)}^2}{\frac{3r^2}{{ϵ}^2}-\frac{1}{{ϵ}^2}-\frac{2}{\Delta t}+(k\pi {)}^2}.$$ (20)

For the 2D case, we choose $f\left(x\right)=\text{cos}\left(k\pi x_1\right)\text{cos}\left(l\pi x_2\right)$ for $k,l\in {\mathbb{N}}^{+}$, and have that $\psi =B\text{cos}\left(k\pi x_1\right)\text{cos}\left(l\pi x_2\right)$, with the coefficient $B$ given by

$$B=-\frac{\frac{3c^2}{{ϵ}^2}-\frac{1}{{ϵ}^2}+\frac{2}{\Delta t}+(k\pi {)}^2+(l\pi {)}^2}{\frac{3r^2}{{ϵ}^2}-\frac{1}{{ϵ}^2}-\frac{2}{\Delta t}+(k\pi {)}^2+(l\pi {)}^2}.$$ (21)

Using $r+\psi \left(x\right)$ as an initial guess, we employ Newton’s method to solve Eq. (10) for ${\varphi }^n$ given ${\varphi }^{n+1}=c+\delta f\left(x\right)$. As an illustrative example, we present the results in Fig. 2 for the 1D case and Fig. 3 for the 2D case with the parameters $c=0.984375,ϵ=0.1$, and $\Delta t=0.01$. We initiate the process with $\delta =0.001$ and employ a homotopy continuation method to compute the solution with $\delta =0.5$ [18, 19, 20]. The solutions of ${\varphi }^n$ corresponding to $\delta =0.5$ for different perturbation modes $k=1$ and $k=5$ are shown in Fig. 2. For the 2D case, perturbation is given as $f\left(x\right)=\text{cos}\left(\pi x_1\right)\text{cos}\left(\pi x_2\right)$ and results are shown in Fig. 3. In this particular instance, we choose $r=\pm 1.99310$ from the interval $\pm \left[r_1,r_2\right]$.

Figure 2:

The solutions of ${\varphi }^n\approx r+\delta B\text{cos}\left(k\pi x_1\right)$ of the CN scheme for ${\varphi }^{n+1}=c+\delta \text{cos}\left(k\pi x_1\right)$ with $\left|\delta \right|=0.5$ are depicted in A and B panels. Here the solid curve is for $k=1$, the dashed curve is for $k=5$. The parameters are chosen as in A $(c=0.984375)$, $(r=-1.99310)$ and B $(c=-0.984375)$, $(r=1.99310)$, $ϵ=0.1$, and $\Delta t=0.01$. These $r$ values are chosen from Table 1 in interval ($r_1,r_2$). In panel C, the CN scheme jumps to a different solution with the initial conditions of ${\varphi }^n$, and ultimately converges to an incorrect solution. Conversely, the backward Euler scheme converges to a solution near ${\varphi }^n$.

Figure 3:

The solutions of ${\varphi }^n\approx r+\delta B\text{cos}\left(k\pi x_1\right)\text{cos}\left(l\pi x_2\right)$ of the CN scheme for 2D function ${\varphi }^{n+1}=c+\delta \text{cos}\left(k\pi x_1\right)\text{cos}\left(l\pi x_2\right)$ with $\left|\delta \right|=0.5$ are depicted in A and B panels. Here the perturbation function is $k=1,l=1$. The parameters are chosen as in A $(c=0.984375)$, $(r=-1.99310)$ and B $(c=-0.984375)$, $(r=1.99310)$, $ϵ=0.1$, and $\Delta t=0.01$. This $r$ values are chosen from Table 1 in interval $\left(r_1,r_2\right)$. In panel C, the CN scheme jumps to a different solution with the initial conditions of ${\varphi }^n$, ultimately converging to an incorrect solution. Conversely, the backward Euler scheme converges to a solution near ${\varphi }^n$.

Consequently, the CN scheme converges after a single time step but yields an incorrect solution. To elaborate, if we choose ${\varphi }^n\left(x\right)\approx 1.99310$ as an initial condition, the CN scheme jumps to approximately −1 after a one-time step and continues to converge toward −1 after a few iterations. Conversely, the backward Euler scheme converges to a correct solution near ${\varphi }^n$. Due to the robustness and stability of the backward Euler scheme, the numerical solution computed using this scheme serves as the reference solution for comparing with solutions computed using other schemes. Moreover, based on the PDE theory, the time evolution solution consistently converges to the nearby steady-state solution. Thus we conclude that the CN scheme converges to an incorrect steady-state solution with this initial condition.

Furthermore, we choose $r=\pm 5.074$ within the interval $\pm \left[r_9,r_{10}\right]$. As a result, the CN scheme converges after 9 zig-zag iterations but leads to an incorrect solution after 14 time steps shown in Fig. 4 for the 1D case and Fig. 5 for the 2D case. The zig-zag curves in Figs. 4 and 5 demonstrate the patterns illustrated in Fig. 1. We choose different initial conditions with both $k=1$ and $k=5$ modes as well as $\delta =0.1$ for the 1D case in Fig. 4 and for the 2D case in Fig. 5.

Figure 4:

The solutions of ${\varphi }^n\approx r+\delta B\text{cos}\left(k\pi x_1\right)$ of the CN scheme for ${\varphi }^{n+1}=c+\delta \text{cos}\left(k\pi x_1\right)$ with $\left|\delta \right|=0.1$ are depicted in A and B panels. Here the solid curve is for $k=1$, the dashed curve is for $k=5$. The parameters are chosen as in A $(c=4.8)$, $(r=-5.074)$ and B $(c=-4.8)$, $(r=5.074)$, $ϵ=0.1$, and $\Delta t=0.01$. In panel C, the CN scheme jumps back and forth, ultimately converging to an incorrect solution. Red and blue regions are visualizing convergence intervals as Fig. 1.

Figure 5:

The solutions of ${\varphi }^n\approx r+\delta B\text{cos}\left(k\pi x_1\right)\text{cos}\left(l\pi x_2\right)$ of the CN scheme for 2D function ${\varphi }^{n+1}=c+\delta \text{cos}\left(k\pi x_1\right)\text{cos}\left(l\pi x_2\right)$ with $\left|\delta \right|=0.1$ are depicted in A and B panels. Here the perturbation function is $k=1,l=1$. The parameters are chosen as in A $(c=4.8)$, $(r=-5.074)$ and B $(c=-4.8)$, $\left(r=5.074\right)ϵ=0.1$, and $\Delta t=0.01$. In panel C, the CN scheme jumps back and forth, ultimately converging to an incorrect solution. Red and blue regions are visualizing convergence intervals as Fig. 1.

5. Convex Splitting of Modified Crank-Nicolson Scheme

Next, we consider the convex splitting of the modified Crank-Nicolson (Mod CN) scheme of Eq. (1) defined as follows:

$$\left\{\begin{array}{l}\frac{{\varphi }^{n+1}-{\varphi }^n}{\Delta t}-\frac{1}{2}\left(\Delta {\varphi }^{n+1}+\Delta {\varphi }^n\right)+\frac{1}{4{ϵ}^2}\left({\varphi }^{n+1}+{\varphi }^n\right)\left({\left({\varphi }^{n+1}\right)}^2+{\left({\varphi }^n\right)}^2\right)-\frac{1}{{ϵ}^2}\left({\varphi }^n\right)=0\\ \frac{\partial {\varphi }^n\left(x\right)}{\partial \mathbf{n}}=\frac{\partial {\varphi }^{n+1}\left(x\right)}{\partial \mathbf{n}}=0,x\in \partial \Omega .\end{array}\right.$$ (22)

5.1. Unconditional stability

We now examine the uniqueness of ${\varphi }^{n+1}=c+\delta \psi \left(x\right)$ for a fixed ${\varphi }^n=r+\delta f\left(x\right)$ within Eq. (22). Substituting these functions into Eq. (22), we obtain:

$$-\frac{1}{2}\Delta \psi +\left[\frac{1}{\Delta t}+\frac{\left.2c^2+(c+r{)}^2\right)}{4{ϵ}^2}\right]\psi +O\left(\delta \right)=G\left(x\right),$$ (23)

where $G\left(x\right)$ is given by:

$$G\left(x\right)=\frac{f\left(x\right)}{\Delta t}+\frac{1}{2}\Delta f\left(x\right)-\frac{1}{4{ϵ}^2}\left(3r^{2}f+c^{2}f+2crf-4f\right).$$ (24)

Given that $\frac{1}{\Delta t}+\frac{2c^2+(c+r{)}^2}{4{ϵ}^2}>0$, it becomes evident that only the trivial zero general solution exists within

$$-\frac{1}{2}\Delta \psi +\left[\frac{1}{\Delta t}+\frac{2c^2+(c+r{)}^2}{4{ϵ}^2}\right]\psi =0.$$ (25)

We conclude that the particular solution for Eq. (23) is unique for any given ${\varphi }^n$, independent of $\Delta t$. This demonstrates the unconditional stability of the convex splitting of the modified Crank-Nicolson scheme.

5.2. The robustness analysis

Similar to the CN scheme, we analyze the trivial solution structure of the convex splitting of the Mod CN scheme firstly. Namely, for ${\varphi }^{n+1}\equiv c$ and ${\varphi }^n\equiv r$, we have

$$\frac{c-r}{\Delta t}+\frac{1}{4{ϵ}^2}(r+c)\left(r^2+c^2\right)-\frac{r}{{ϵ}^2}=0.$$ (26)

Through a computation similar to the one used in analyzing the CN scheme §4.2.1, we can calculate the sequence of $r_n$ by solving Eq. (26) with $c=r_{n-1}$ (here $r_0=0$). We present the values of $r_n$ for various ratios of $\frac{\Delta t}{2{ϵ}^2}$ in Table 2.

Table 2:

Convergence Interval points $r_i$ for the Convex splitting of modified CN scheme with different $\frac{\Delta t}{2{ϵ}^2}$.

$\frac{\Delta t}{2{ϵ}^2}$	$r_1$	$r_2$	$r_3$	$r_4$
0.001	44.766	75.889	98.476	116.931
0.01	14.283	24.165	31.334	37.192
0.1	4.899	8.147	10.497	12.418
0.25	3.464	5.641	7.212	8.497
0.5	2.828	4.503	5.707	6.694

We can perturb the trivial solutions using ${\varphi }^{n+1}=c+\delta f\left(x\right)$ and ${\varphi }^n=r+\delta \psi \left(x\right)$ in a way similar to the one in §4.2.2. This allows us to reformulate the Convex splitting of Mod CN scheme as follows:

$$-\frac{1}{2}\Delta \psi +\left(-\frac{1}{\Delta t}+\frac{3r^2+c^2+2cr-4}{4{ϵ}^2}\right)\psi =G\left(x\right),$$ (27)

where $G\left(x\right)$ is given by:

$$G\left(x\right)=-\frac{f\left(x\right)}{\Delta t}+\frac{1}{2}\Delta f\left(x\right)-\frac{1}{4{ϵ}^2}\left(3c^{2}f+r^{2}f+2crf\right).$$ (28)

When we choose $f\left(x\right)=\text{cos}\left(k\pi x_1\right)$, the particular solution becomes $\psi =B\text{cos}\left(k\pi x_1\right)$ for the 1D case, where the coefficient $B$ is determined by the following expression from using Eq. (27):

$$B=-\frac{\frac{1}{\Delta t}+\frac{k^2{\pi }^2}{2}+\frac{3c^2+r^2+2cr}{4{ϵ}^2}}{-\frac{1}{\Delta t}+\frac{3r^2+c^2+2cr-4}{4{ϵ}^2}+\frac{k^2{\pi }^2}{2}}=-\frac{\frac{3c^2+r^2+2cr}{2{ϵ}^2}+\frac{2}{\Delta t}+k^2{\pi }^2}{\frac{3r^2+c^2+2cr-4}{2{ϵ}^2}-\frac{2}{\Delta t}+k^2{\pi }^2}.$$ (29)

Using the computed $\psi \left(x\right)$ as an initial approximation, we employ Newton’s method to solve for ${\varphi }^n$ given ${\varphi }^{n+1}=c+\delta f\left(x\right)$ in Eq. (22). This results in a solution structure similar to that of the CN scheme.

For 2D we have similar results as $f\left(x\right)=\text{cos}\left(k\pi x_1\right)\text{cos}\left(l\pi x_2\right)$ for $k,l\in {\mathbb{N}}^{+}$. We find that $\psi =B\text{cos}\left(k\pi x_1\right)\text{cos}\left(l\pi x_2\right)$, with the coefficient $B$ given by

$$B=-\frac{\frac{3c^2+r^2+2cr}{2{ϵ}^2}+\frac{2}{\Delta t}+k^2{\pi }^2+l^2{\pi }^2}{\frac{3r^2+c^2+2cr-4}{2{ϵ}^2}-\frac{2}{\Delta t}+k^2{\pi }^2+l^2{\pi }^2}.$$ (30)

6. Diagonally Implicit Runge–Kutta (DIRK)

The DIRK family of methods is the most widely used implicit Runge-Kutta (IRK) method for solving phase field modeling problems due to their relative ease of implementation [26]. Some applications of the DIRK methods on the Allen-Cahn equation can be found in [7, 34]. These methods are characterized by a lower triangular A-matrix with at least one non-zero diagonal entry and are sometimes referred to as semi-implicit or semi-explicit Runge-Kutta methods. This structure allows for solving each stage individually rather than all stages simultaneously. We write the general formula of DIRK method in Butcher array format as follows:

When solving the Allen-Cahn equation, the DIRK method is summarized as

$${\varphi }^{n+1}={\varphi }^n+\Delta t\sum _{i=1}^s{b}_i{k}_i,$$ (31)

where

$$k_i=F\left({\varphi }^n+\Delta t\sum _{j=1}^i{a}_{ij}{k}_j\right)\text{and}F\left(\varphi \right)=\Delta \varphi -\frac{1}{{ϵ}^2}\left({\varphi }^3-\varphi \right).$$ (32)

By letting ${\varphi }_i^{n+1}={\varphi }^n+\Delta t{\sum }_{j=1}^i{a}_{ij}{k}_j$, we can rewrite the DIRK method as [26, 27]

$${\varphi }_i^{n+1}=\left\{\begin{array}{ll}{\varphi }^n& \text{for}i=0,\\ {\varphi }_0^{n+1}+\Delta t\sum _{j=1}^i{a}_{ij}F\left({\varphi }_j^{n+1}\right)& \text{for}1\le i\le s.\end{array}\right.$$ (33)

In this case, the final solution can be expressed as ${\varphi }^{n+1}={\varphi }_0^{n+1}+\Delta t{\sum }_{i=1}^s{b}_{i}F\left({\varphi }_i^{n+1}\right)$. Then the DIRK method represents a multi-stage backward Euler method which solves for ${\varphi }_i^{n+1}$ for each stage $i$.

6.1. The stability condition via bifurcation analysis

First, we consider a trivial solution case of ${\varphi }_{i+1}^{n+1}\equiv c$ for given ${\varphi }_0^{n+1},{\varphi }_1^{n+1},\dots ,{\varphi }_{i-1}^{n+1}$, specifically, ${\varphi }_i^{n+1}\equiv r$:

$${\varphi }_{i+1}^{n+1}={\varphi }_0^{n+1}+\Delta t\sum _{j=1}^{i+1}{a}_{i+1,j}F\left({\varphi }_j^{n+1}\right)$$ (34)

Next, let’s perturb the trivial solutions with ${\varphi }_{i+1}^{n+1}\left(x\right)=c+\delta \psi \left(x\right)$ and ${\varphi }_i^{n+1}\left(x\right)=r+\delta f\left(x\right)$. By substituting into Eq. (34) and retaining the linear term in $\delta$, we obtain

$$-\Delta \psi \left(x\right)+\left(\frac{1}{\Delta t{a}_{i+1,i+1}}+\frac{3c^2-1}{{ϵ}^2}\right)\psi \left(x\right)=\left\{\begin{array}{ll}\frac{f\left(x\right)}{a_{\mathrm{1,1}}}+O\left(\delta \right)& \text{if}i=0\\ \frac{a_{i+1,i}}{a_{i+1,i+1}}\left(\Delta f-\frac{1}{{ϵ}^2}\left(3r^{2}f-f\right)\right)+O\left(\delta \right)& \text{else}\end{array}\right.$$ (35)

We aim to assess the uniqueness of ${\varphi }_{i+1}^{n+1}$ while holding ${\varphi }_i^{n+1}$ fixed. In this case, $f\left(x\right)$ is a given function, and we focus on the uniqueness of the homogeneous solution, namely,

$$-\Delta \psi \left(x\right)+\left(\frac{1}{\Delta t{a}_{i+1,i+1}}+\frac{3c^2-1}{{ϵ}^2}\right)\psi \left(x\right)=0.$$ (36)

This is essentially the same as Eq. (5) if $a_{i+1,i+1}=1$. By Proposition 3.1, we have that the solution is unique if

$$\left(\frac{1}{\Delta t{a}_{i+1i+1}}+\frac{3c^2-1}{{ϵ}^2}\right)\ge 0,$$ (37)

or

$$0\le \Delta t{a}_{i+1i+1}\le {ϵ}^2.$$ (38)

Consequently, we can conclude that the particular solution for Eq. (34) is unique, and ${\varphi }_{i+1}^{n+1}$ is also unique while satisfying the stability condition $\Delta t{a}_{i+1i+1}\le {ϵ}^2$. Then the stability condition of the DIRK method is

$$0\le \Delta t\le \frac{{ϵ}^2}{{\text{max}}_i{a}_{ii}}.$$ (39)

6.2. The robustness analysis

In theory, we can apply robustness analysis to any order of the DIRK method. In this section, for simplicity, we illustrate the idea by considering the 2nd order DIRK method with the following 2 × 2 Butcher array:

We first analyze the trivial solution case, namely ${\varphi }^{n+1}\equiv c$ and ${\varphi }^n\equiv r$. Then, the DIRK method with 2nd order for solving the Allen-Cahn equation is expressed as:

$$\left\{\begin{array}{l}{\varphi }_1^{n+1}={\varphi }_0^{n+1}+\frac{\Delta {\varphi }_1^{n+1}}{4}-\frac{\Delta t}{4{ϵ}^2}\left({\left({\varphi }_1^{n+1}\right)}^3-{\varphi }_1^{n+1}\right),\\ {\varphi }_2^{n+1}={\varphi }_1^{n+1}+\frac{\Delta {\varphi }_1^{n+1}}{4}-\frac{\Delta t}{4{ϵ}^2}\left({\left({\varphi }_1^{n+1}\right)}^3-{\varphi }_1^{n+1}\right)+\frac{\Delta {\varphi }_2^{n+1}}{4}-\frac{\Delta t}{4{ϵ}^2}\left({\left({\varphi }_2^{n+1}\right)}^3-{\varphi }_2^{n+1}\right),\\ {\varphi }^{n+1}={\varphi }_2^{n+1}+\frac{\Delta {\varphi }_2^{n+1}}{4}-\frac{\Delta t}{4{ϵ}^2}\left({\left({\varphi }_2^{n+1}\right)}^3-{\varphi }_2^{n+1}\right).\end{array}\right.$$ (40)

By letting ${\varphi }^{n+1}\equiv 0$, we have 3 roots for ${\varphi }_2^{n+1}$ as $\pm \frac{r_1}{2}$ and 0 by solving the last equation in Eq. (40), where $r_1=2\sqrt{1+\frac{4{ϵ}^2}{\Delta t}}$. By simplifying Eq. (40) with ${\varphi }^{n+1}\equiv 0$, we have

$$2{\varphi }_1^{n+1}-2{\varphi }_2^{n+1}={\varphi }_0^{n+1}.$$ (41)

By plugging Eq. (41) into the first equation of Eq. (40), we have

$$\frac{-{\varphi }_0^{n+1}+2{\varphi }_2^{n+1}}{2}=-\frac{\Delta t}{4{ϵ}^2}\left({\left(\frac{{\varphi }_0^{n+1}+2{\varphi }_2^{n+1}}{2}\right)}^3-\frac{{\varphi }_0^{n+1}+2{\varphi }_2^{n+1}}{2}\right).$$ (42)

If ${\varphi }_2^{n+1}=\pm \frac{r_1}{2}$, since of the discriminant of the cubic polynomial from the second equation in Eq. (40),

$$-27{\left(\frac{8{ϵ}^2}{\Delta t}{\varphi }_2^{n+1}\right)}^2+4{\left(\frac{4{ϵ}^2}{\Delta t}+1\right)}^3<0,\text{when}\Delta t<4{ϵ}^2,$$ (43)

we have only one root as ${\varphi }_1^{n+1}=\pm h_1$. Thus we have

$${\varphi }_2^{n+1}=\pm \frac{r_1}{2},{\varphi }_1^{n+1}=\pm \frac{-s_1+2{\varphi }_2^{n+1}}{2},\text{and}{\varphi }_0^{n+1}=\mp s_1,$$ (44)

where $s_1$ is the root of

$$\frac{s_1+r_1}{2}=-\frac{\Delta t}{4{ϵ}^2}\left({\left(\frac{-s_1+r_1}{2}\right)}^3-\frac{-s_1+r_1}{2}\right).$$ (45)

If ${\varphi }_2^{n+1}=0$, we have another set of solutions:

$${\varphi }_2^{n+1}=0,{\varphi }_1^{n+1}=\pm \frac{r_1}{2}\text{or}0,{\varphi }_0^{n+1}=2{\varphi }_1^{n+1}-2{\varphi }_2^{n+1}.$$ (46)

For $0<r_1<s_1$, we have five roots for ${\varphi }_0^{n+1}$, namely, ${\varphi }_0^{n+1}=\pm r_1,\mp s_1$, and 0.

By letting ${\varphi }^{n+1}=s_1$, we obtain a unique solution for ${\varphi }^n$ due to the discriminant of the cubic polynomial, denoted as ${\varphi }^n=s_2$. Inductively, we can define $s_i$ and $r_i$ by solving Eq. (40) for ${\varphi }^n$ with ${\varphi }^{n+1}=s_{i-1}$ and ${\varphi }^{n+1}=r_{i-1}$, respectively. The values of $r_i$ and $s_i$ for different values of $\frac{\Delta t}{4{ϵ}^2}$ are shown in Table 3, and the iterations of the DIRK method in different regions are illustrated in Fig. 6.

Table 3:

Convergence Interval points $r_i,s_i$ for the DIRK 2nd order scheme in Eq. (40) with different $\frac{\Delta t}{4{ϵ}^2}$.

$\frac{\Delta t}{4{ϵ}^2}$	$r_1$	$s_1$	$r_2$	$s_2$	$r_3$	$s_3$	$r_4$	$s_4$
0.001	63.277	159.524	280.251	421.311	580.137	754.936	944.371	1147.391
0.01	20.1	50.612	88.857	133.527	183.81	239.141	299.098	363.349
0.1	6.633	16.517	28.821	43.14	59.221	76.889	96.012	116.485
0.25	4.472	10.958	18.95	28.2	38.552	49.898	62.156	75.262
0.5	3.464	8.306	14.188	20.942	28.462	36.675	45.524	54.966

Figure 6:

Visualizing Convergence Intervals of DIRK scheme with 2nd order in Eq. (40). Suppose initial conditions are chosen in red regions. In that case, DIRK with 2nd order scheme eventually converges to 1, while the initial conditions chosen in blue regions lead the DIRK method to converge to −1. The values of $r_i,s_i$ with different $\frac{\Delta t}{4{ϵ}^2}$ are shown in Table 3.

Next, we perturb the trivial solutions using ${\varphi }^{n+1}=c+\delta \xi \left(x\right),{\varphi }_2^{n+1}=c_2+\delta {\xi }_2\left(x\right),{\varphi }_1^{n+1}=c_1+\delta {\xi }_1\left(x\right)$ and ${\varphi }_0^{n+1}=r+\delta {\xi }_0\left(x\right)$ as similar way in §4.2.2, where $\xi \left(x\right)$ is a given perturbed function. After plugging in to Eq. (40) and retaining the linear term in $\delta$, we have

$$\left\{\begin{array}{l}{\xi }_1={\xi }_0+\frac{\Delta t}{4}\Delta {\xi }_1-\frac{\Delta t}{4{ϵ}^2}\left(3{\xi }_1{\left(c_1\right)}^2-{\xi }_1\right)+O\left(\delta \right),\\ {\xi }_2={\xi }_1+\frac{\Delta t}{4}\Delta {\xi }_2+\frac{\Delta t}{4}\Delta {\xi }_1-\frac{\Delta t}{4{ϵ}^2}\left(3{\xi }_1{\left(c_1\right)}^2-{\xi }_1\right)-\frac{\Delta t}{4{ϵ}^2}\left(3{\xi }_2{\left(c_2\right)}^2-{\xi }_2\right)+O\left(\delta \right),\\ \xi \left(x\right)={\xi }_2+\frac{\Delta t}{4}\Delta {\xi }_2-\frac{\Delta t}{4{ϵ}^2}\left(3{\xi }_2{\left(c_2\right)}^2-{\xi }_2\right)+O\left(\delta \right).\end{array}\right.$$ (47)

By choosing specific functions in the 1D case as

$$\xi \left(x\right)=\text{cos}\left(k\pi x_1\right),{\xi }_2\left(x\right)=B_2\text{cos}\left(k\pi x_1\right),{\xi }_1\left(x\right)=B_1\text{cos}\left(k\pi x_1\right),{\xi }_0\left(x\right)=B_0\text{cos}\left(k\pi x_1\right)$$ (48)

we obtain

$$\{\begin{array}{l}{B}_2=\frac{1}{1-\frac{\Delta t{k}^2{\pi }^2}{4}-\frac{\Delta t}{4{ϵ}^2}\left(3{\left(c_2\right)}^2-1\right)}\hfill \\ B_1=B_2\frac{1-\frac{\Delta t{k}^2{\pi }^2}{4}+\frac{\Delta t}{4{ϵ}^2}\left(3{\left(c_2\right)}^2-1\right)}{1-\frac{\Delta t{k}^2{\pi }^2}{4}-\frac{\Delta t}{4{ϵ}^2}\left(3{\left(c_1\right)}^2-1\right)}\hfill \\ B_0=B_1\left(1+\frac{\Delta t{k}^2{\pi }^2}{4}+\frac{\Delta t}{4{ϵ}^2}\left(3{\left(c_1\right)}^2-1\right)\right)\hfill \end{array}$$ (49)

Then, we employ Newton’s method to solve Eq. (40) for ${\varphi }^n$, given ${\varphi }^{n+1}=c+\delta \xi \left(x\right)$, taking the initial guess ${\varphi }^n\approx r+\delta B_0\text{cos}\left(k\pi x_1\right)$. As an illustrative example, we show the solutions of ${\varphi }^n$ in Fig. 7 for the 1D case and Fig. 8 for the 2D case with the parameters $c=-7,ϵ=0.1$, and $\Delta t=0.01$. We initiate the process with $\delta =0.001$ and employ a homotopy continuation method to compute the solution with $\delta =0.1$ [18, 19, 20]. The solutions of ${\varphi }^n$ corresponding to $\delta =0.1$ for different perturbation modes $k=1$ and $k=5$ are shown in Fig. 7. For the 2D case, perturbation is given as $\xi \left(x\right)=\text{cos}\left(\pi x_1\right)\text{cos}\left(\pi x_2\right)$ and we present the results in Fig. 8.

Figure 7:

The solutions of ${\varphi }^n\approx r+\delta B_0\text{cos}\left(k\pi x_1\right)$ of the DIRK scheme for ${\varphi }^{n+1}=c+\delta \text{cos}\left(k\pi x_1\right)$ with $\left|\delta \right|=0.1$ are depicted in A and B panels. Here the solid curve is for $k=1$, the dashed curve is for $k=5$. The parameters are chosen as in A $(c=-7)$, $(r=-22.70665)$ and B $(c=7)$, $(r=22.70665)$, $ϵ=0.1$, and $\Delta t=0.01$. This $c$ values are chosen from Table 3 in interval $\left(r_1,s_1\right)$. In panel C, the DIRK scheme jumps to a different solution with the initial conditions of ${\varphi }^n$, ultimately converging to an incorrect solution. Conversely, the backward Euler scheme converges to a solution near ${\varphi }^n$.

Figure 8:

The solutions of ${\varphi }^n\approx r+\delta B_0\text{cos}\left(k\pi x_1\right)\text{cos}\left(l\pi x_2\right)$ of the DIRK scheme for 2D function ${\varphi }^{n+1}=c+\delta \text{cos}\left(k\pi x_1\right)\text{cos}\left(l\pi x_2\right)$ with $\left|\delta \right|=0.1$ are depicted in A and B panels. Here the perturbation function is $k=1,l=1$. The parameters are chosen as in A $(c=-7)$, $(r=-22.70665)$ and B $(c=7)$, $(r=22.70665)$, $ϵ=0.1$, and $\Delta t=0.01$. This $c$ values are chosen from Table 3 in interval $\left(r_1,s_1\right)$. In panel C, the DIRK scheme jumps to a different solution with the initial conditions of ${\varphi }^n$, ultimately converging to an incorrect solution. Conversely, the backward Euler scheme converges to a solution near ${\varphi }^n$.

Thus we use ${\varphi }^n$ as the initial condition to solve the Allen-Cahn equation by using DIRK with second order. Since we choose $c=\pm 7$ from the interval $\pm \left[r_1,s_1\right]$, consequently, the DIRK scheme converges after two time steps but yields an incorrect solution. To elaborate, if we choose ${\varphi }^{n+1}\left(x\right)\approx 7$ as an initial condition, the DIRK scheme jumps to approximately −1 after a one-time step and continues to converge toward −1 after a few iterations. Conversely, the backward Euler scheme converges to a correct solution near ${\varphi }^n$.

Furthermore, we choose $r=\pm 95.72$ within the interval $\pm \left[r_5,s_5\right]$. As a result, the DIRK scheme converges after five iterations but leads to an incorrect solution. This convergence process is evident in the jumping curves depicted in Fig. 6. The initial conditions for $k=1$ and $k=5$ modes with $\delta =0.1$, as well as the final solutions after 7 time steps, are shown in Fig. 9 for the 1D case and Fig. 10 for the 2D case.

Figure 9:

The solutions of ${\varphi }^n\approx r+\delta B_0\text{cos}\left(k\pi x_1\right)$ of the DIRK scheme for function ${\varphi }^{n+1}=c+\delta \text{cos}\left(k\pi x_1\right)$ with $\left|\delta \right|=0.1$ are depicted in A and B panels. Here the solid curve is for $k=1$, the dashed curve is for $k=5$. The parameters are chosen as in A $(c=-68)$, $(r=-95.72)$ and B $(c=68)$, $(r=95.72)$, $ϵ=0.1$, and $\Delta t=0.01$. In panel C, the DIRK scheme, ultimately converging to an incorrect solution. Red and blue regions are visualizing convergence intervals as Fig. 6.

Figure 10:

The solutions of ${\varphi }^n\approx r+\delta B_0\text{cos}\left(k\pi x_1\right)\text{cos}\left(l\pi x_2\right)$ of the DIRK scheme for 2D function ${\varphi }^{n+1}=c+\delta \text{cos}\left(k\pi x_1\right)\text{cos}\left(l\pi x_2\right)$ with $\left|\delta \right|=0.1$ are depicted in A and B panels. Here the perturbation function is $k=1,l=1$. The parameters are chosen as in A $(c=-68)$, $(r=-95.72)$ and B $(c=68)$, $(r=95.72)$, $ϵ=0.1$, and $\Delta t=0.01$. In panel C, the DIRK scheme converges to an incorrect solution. Red and blue regions visualize convergence intervals as Fig. 6.

7. Benchmark problem test

Numerous benchmark problems have been utilized for Allen-Cahn equation, with numerical results validated through various computational methods employing different spatial and temporal discretizations [7]. In this section, we consider two benchmark problems from [7]. The first benchmark problem uses the following initial condition:

$$u\left(x,y,0\right)=\text{tanh}\left(\frac{\sqrt{(x-\pi {)}^2+(y-\pi {)}^2}-2}{ϵ\sqrt{2}}\right)+c,$$ (50)

on the domain $[0,2\pi {]}^2$, where $c$ is a constant used to test the robustness of different numerical schemes. First, we performed 10 iterations of the CN scheme with a time step $\Delta t=0.002$ and parameter $ϵ=0.1$. The results, shown in Fig. 11, indicate that the CN scheme lacks robustness, as positive initial conditions converge to the negative steady state. Next, we test the benchmark problem using the convex splitting of modified CN scheme in Eq. (22) and the second-order RK method in Eq. (40). The results, shown in Fig. 12, similarly indicate the lack of robustness in both schemes.

Figure 11:

The first benchmark problem using the CN scheme with the initial conditions given by Eq. (50). The top-left figure corresponds to $c=5.5$ and the top-right figure to $c=7.5$. The middle row displays the numerical results after applying the CN scheme for 10 time steps. The bottom figure illustrates the dynamics of the average solution over the domain as a function of the time steps.

Figure 12:

The first benchmark problem using the convex splitting of modified CN scheme and 2nd order RK method with the initial conditions given by Eq. (50). We show the dynamics of the average solution over the domain as a function of the time steps with different constant $c$.

The second benchmark problem we considered uses the following initial conditions [7]:

$$u(x,y,0)=c+\tilde{\varphi }\left(\sqrt{{\left(x-x_i\right)}^2+{\left(y-y_i\right)}^2}-r_i\right)\text{and}\tilde{\varphi }\left(s\right)=\left\{\begin{array}{ll}2e^{-{ϵ}^2/s^2},& \text{if}s<0\\ 0,& \text{otherwise}\end{array}\right.$$ (51)

on the domain $[\mathrm{0,2}\pi {]}^2$ with coefficients listed in Table 4.

Table 4:

Centres $\left(x_i,y_i\right)$ and radii $r_i$ for the initial conditions in Eq. (51).

$i$	1	2	3	4	5	6	7
$x_i$	$\frac{\pi }{2}$	$\frac{\pi }{4}$	$\frac{\pi }{2}$	$\pi$	$\frac{3\pi }{2}$	$\pi$	$\frac{3\pi }{2}$
$y_i$	$\frac{\pi }{2}$	$\frac{3\pi }{4}$	$\frac{5\pi }{4}$	$\frac{\pi }{4}$	$\frac{\pi }{4}$	$\pi$	$\frac{3\pi }{2}$
$r_i$	$\frac{\pi }{5}$	$\frac{2\pi }{15}$	$\frac{2\pi }{15}$	$\frac{\pi }{10}$	$\frac{\pi }{10}$	$\frac{\pi }{4}$	$\frac{\pi }{4}$

Similarly, we tested all three schemes—CN, convex splitting of modified CN scheme, and the second-order RK method—with a time step of $\Delta t=0.002$ and parameter $ϵ=0.1$. The results, presented in Figs. 13 and 14, indicate a lack of robustness in each of the schemes.

Figure 13:

The second benchmark problem using the CN scheme with the initial conditions given by Eq. (51). The top-left figure corresponds to $c=5.5$ and the top-right figure to $c=7.5$. The middle row displays the numerical results after applying the CN scheme for 10 time steps. The bottom figure illustrates the dynamics of the average solution over the domain as a function of the time steps.

Figure 14:

The first benchmark problem using the convex splitting of modified CN scheme and 2nd order RK method with the initial conditions given by Eq. (51). We show the dynamics of the average solution over the domain as a function of the time steps with different constant $c$.

8. Conclusions

The Allen-Cahn equation serving as a fundamental tool for modeling phase transitions, offers invaluable insights into interface evolution across diverse physical systems. In this paper, we have devoted into the stability and robustness of various time-discretization numerical schemes utilized to solve the Allen-Cahn equation, recognizing their pivotal role in ensuring precise simulations in practical applications.

Our stability analyses of several numerical methods, including the backward Euler, Crank-Nicolson, Convex Splitting of modified Crank-Nicolson schemes, and the DIRK method, have unveiled fundamental stability conditions for each method. Notably, the backward Euler scheme, Crank-Nicolson, and DIRK methods exhibited conditional stability, necessitating careful consideration of time step sizes. Conversely, the convex splitting of the modified Crank-Nicolson scheme showcased unconditional stability, affording flexibility in time step selection without compromising numerical accuracy.

Furthermore, our robustness analyses have shed light on the behavior of numerical solutions under varying initial conditions. While the backward Euler method demonstrated robustness, reliably converging to physical solutions regardless of initial conditions; other methods such as the Crank-Nicolson and convex splitting of modified Crank-Nicolson schemes, as well as the DIRK method, exhibited sensitivity to initial conditions in the solving of these nonlinear schemes at each time step, potentially leading to wrong solutions if the initial conditions are not carefully chosen.

To conclude, our study introduces the concepts of stability and robustness to the realm of numerical methods for solving the Allen-Cahn equation. By elucidating the stability conditions and robustness characteristics of these methods, we provide a novel framework for evaluating numerical techniques tailored to nonlinear differential equations, thereby advancing the accuracy and reliability of phase transition simulations in various scientific domains.

Table 5:

Summary of stability conditions for different numerical schemes. $a_{ii}$ in DIRK is the diagonal elements in Butcher array format.

Numerical Scheme	Backward Euler	CN	Convex Splitting of Modified CN	DIRK
Stability Condition	$\Delta t\le {ϵ}^2$	$\Delta t\le 2{ϵ}^2$	$\Delta t\le \mathrm{\infty }$	$\Delta t\le \frac{{ϵ}^2}{{max}_i{a}_{ii}}$

Highlights.

• Comprehensive stability and robustness analysis of numerical schemes for the Allen-Cahn equation.

• Backward Euler method shows robust convergence to correct physical solutions regardless of initial conditions.

• Sensitivity to initial conditions highlighted for Crank-Nicolson and Diagonally Implicit Runge-Kutta (DIRK) methods, which may lead to incorrect solutions.