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

TL;DR

The paper addresses how time-discretization schemes for the Allen-Cahn equation perform in terms of stability and robustness, introducing a bifurcation- and perturbation-based framework. It analyzes BE, CN, Mod CN with convex splitting, and DIRK, deriving explicit time-step constraints and revealing that only the unconditional Mod CN scheme guarantees stability irrespective of $\Delta t$. The findings show BE is robust to initial guesses, while CN and DIRK can converge to incorrect steady states under certain initial data, highlighting the need for careful scheme selection in nonlinear phase-field simulations. Overall, the work provides rigorous criteria for stability and robustness that can guide the design and evaluation of numerical methods for nonlinear PDEs beyond the Allen-Cahn equation.

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 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, the 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.

Figures (10)

• Figure 1: Visualizing Convergence Intervals of CN Scheme in Eq. \ref{['cn scheme']}. 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 $\frac{\Delta t}{2 \epsilon^2}$ are shown in Table \ref{['Table_CN']}.
• Figure 1: Visualizing Convergence Intervals of DIRK scheme with 2nd order in Eq. \ref{['DIRK2']}. 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 $\frac{\Delta t}{4 \epsilon^2}$ are shown in Table \ref{['Table_rk']}.
• Figure 2: The solutions of $\phi^{n} \approx r + \delta B\cos(k\pi x_1)$ of the CN scheme for $\phi^{n+1}=c+\delta \cos(k\pi x_1)$ with $|\delta|=0.5$ 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 $(c=-0.984375)$, ($r=1.99310$), $\epsilon=0.1$, and $\Delta t=0.01$. These $r$ values are chosen from Table \ref{['Table_CN']} in interval $(r_{1},r_{2})$. In panel C, the CN scheme jumps to a different solution with the initial conditions of $\phi^n$, and ultimately converges to an incorrect solution. Conversely, the backward Euler scheme converges to a solution near $\phi^n$.
• Figure 2: The solutions of $\phi^{n} \approx r + \delta B_0\cos(k\pi x_1)$ of the DIRK scheme for $\phi^{n+1}=c+\delta \cos(k\pi x_1)$ with $|\delta|=0.1$ 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 $(c=7)$, ($r=22.70665$), $\epsilon=0.1$, and $\Delta t=0.01$. This $c$ values are chosen from Table \ref{['Table_rk']} in interval $(r_{1},s_{1})$. In panel C, the DIRK scheme jumps to a different solution with the initial conditions of $\phi^n$, ultimately converging to an incorrect solution. Conversely, the backward Euler scheme converges to a solution near $\phi^n$.
• Figure 3: The solutions of $\phi^{n} \approx r + \delta B\cos(k\pi x_1)\cos(l\pi x_2)$ of the CN scheme for 2D function $\phi^{n+1}=c+\delta \cos(k\pi x_1)\cos(l\pi x_2)$ with $|\delta|=0.5$ 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 $(c=-0.984375)$, ($r=1.99310$), $\epsilon=0.1$, and $\Delta t=0.01$. This $r$ values are chosen from Table \ref{['Table_CN']} in interval $(r_{1},r_{2})$. In panel C, the CN scheme jumps to a different solution with the initial conditions of $\phi^n$, ultimately converging to an incorrect solution. Conversely, the backward Euler scheme converges to a solution near $\phi^n$.

Theorems & Definitions (6)

• Definition 1
• Definition 2
• Proposition 1
• Proof 1
• Remark 3.1
• Proposition 1