Title: Applications of Semi-Implicit Fourier-Spectral Method to Phase
        Field Equations

Authors: L. Q. Chen and Jie Shen

Status: To appear in Comput. Phys. Comm.

Abstract:

An efficient and accurate numerical method is implemented for solving
the time-dependent Ginzburg-Landau equation and the Cahn-Hilliard
equation.  The time variable is discretized by using semi-implicit
schemes which allow much larger time step sizes than explicit schemes;
the space variables are discretized by using a Fourier-spectral method
whose convergence rate is exponential in contrast
 to second-order by a usual finite-difference method. We have applied
our method to predict the equilibrium profiles of an order parameter
across a stationary planar interface and the velocity of a moving
interface by solving the time-dependent Ginzburg-Landau equation, and
compared the accuracy and efficiency of our results
 with those obtained by others. We demonstrate that, for a specified
accuracy of $0.5\%$, the speed-up of using semi-implicit
Fourier-spectral method, when compared with the explicit
finite-difference schemes,
is at least two orders of magnitude in two dimensions, and close to
three orders of magnitude in three dimensions. The method is shown to
be particularly powerful for systems in which the morphologies and
microstructures are dominated by long-range elastic interactions.