  Phase-field models are widely employed to simulate microstructure evolution
during processes such as solidification or heat treatment. The resulting
partial differential equations, often strongly coupled together, may be solved
by a broad range of numerical methods, but this often results in a high
computational cost, which calls for advanced numerical methods to accelerate
their resolution. Here, we quantitatively test the efficiency and accuracy of
semi-implicit Fourier spectral-based methods, implemented in Python programming
language and parallelized on a graphics processing unit (GPU), for solving a
phase-field model coupling Cahn-Hilliard and Allen-Cahn equations. We compare
computational performance and accuracy with a standard explicit finite
difference (FD) implementation with similar GPU parallelization on the same
hardware. For a similar spatial discretization, the semi-implicit Fourier
spectral (FS) solvers outperform the FD resolution as soon as the time step can
be taken 5 to 6 times higher than afforded for the stability of the FD scheme.
The accuracy of the FS methods also remains excellent even for coarse grids,
while that of FD deteriorates significantly. Therefore, for an equivalent level
of accuracy, semi-implicit FS methods severely outperform explicit FD, by up to
4 orders of magnitude, as they allow much coarser spatial and temporal
discretization.
