  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.
