Figure 4. The illustration presents the...

Figure 4.

Two plots show spatial error profiles of phi n minus the reference solution at time t n for u zero tau over epsilon equal to 1 and 1 divided by 4, comparing Implicit Euler, Crank-Nicolson, and Generalized alpha methods.

The illustration presents the spatial distribution of the error, defined as the difference between the numerical phase field phi n at time t n and a reference solution phi reference, plotted as a function of the spatial coordinate x. Two cases are shown: u zero times tau divided by epsilon equal to 1 and u zero times tau divided by epsilon equal to 1 divided by 4, where u zero is a characteristic velocity, tau is the time step, and epsilon is the interface thickness parameter. The results compare three time integration schemes: Implicit Euler, Crank-Nicolson, and the Generalized alpha method. For the larger ratio, pronounced oscillations and larger error magnitudes appear near the interface for Implicit Euler, while Crank-Nicolson and especially the Generalized alpha method show reduced and more localized errors. For the smaller ratio, all methods exhibit significantly smaller errors, with the Generalized alpha method producing the smoothest and lowest-amplitude error profile, indicating improved accuracy and numerical stability as u zero tau over epsilon decreases.

Error in the phase-field approximation for the implicit-Euler method, the Crank-Nicolson method, and the third-order generalized- $α$ method at time $u_{0} t_{n} / ε = 48$ for time-step sizes $u_{0} τ / ε = 1$ (left) and $u_{0} τ / ε = \frac{1}{4}$ (right)

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