Figure 1. Two side-by-side graphs of...

Figure 1.

Two plots compare successive tangent-hyperbolic profiles phi n minus 1 and phi n versus x, showing how their separation changes with the ratio u times tau divided by epsilon in different regimes.

Two side-by-side graphs of sigmoid-shaped functions, labeled phi n minus 1, represented by a light curve, and phi n, represented by a dark curve, plotted against the horizontal variable x with values ranging from minus 1 to 1 on the vertical axis. In the left panel, the ratio u times tau divided by epsilon equals 2 times the square root of 2, producing a large vertical separation of about 2 between the two profiles. In the right panel, where u times tau divided by epsilon is much less than 1, the curves are closer together, and the shift between them is of order u times tau divided by epsilon, illustrating how the parameter ratio controls the spacing and transition behavior of successive profiles.

Illustration of the error in the initial estimate of the phase field for Newton’s method obtained from the previous time step for $u τ / ε = 2 \sqrt{2}$ (left) and for $u τ / ε ≪ 1$ (right)

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