Applications of semi-implicit Fourier-spectral method to phase field equations

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 speedup 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.