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

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.