  We study numerically the one-dimensional Allen-Cahn equation with the
spectral fractional Laplacian $(-\Delta)^{\alpha/2}$ on intervals with
homogeneous Neumann boundary conditions. In particular, we are interested in
the speed of sharp interfaces approaching and annihilating each other. This
process is known to be exponentially slow in the case of the classical
Laplacian. Here we investigate how the width and speed of the interfaces change
if we vary the exponent $\alpha$ of the fractional Laplacian. For the
associated model on the real-line we derive asymptotic formulas for the
interface speed and time-to-collision in terms of $\alpha$ and a scaling
parameter $\varepsilon$. We use a numerical approach via a finite-element
method based upon extending the fractional Laplacian to a cylinder in the
upper-half plane, and compute the interface speed, time-to-collapse and
interface width for $\alpha\in(0.2,2]$. A comparison shows that the asymptotic
formulas for the interface speed and time-to-collision give a good
approximation for large intervals.
