On the unconditionally gradient stable scheme for the Cahn-Hilliard equation and its implementation with Fourier method

We implement a nonlinear unconditionally gradient stable scheme by Eyre, within the Fourier method framework for the long-time numerical integration of the Allen-Cahn and CahnHilliard equations, which are gradient flows of the Allen-Cahn energy. We propose a new iterative procedure to solve the nonlinear scheme. When the iterative scheme is applied to the Allen-Cahn equation, we show that the nonlinear iteration is a contractive mapping in the L2 norm for large time steps. For the Cahn-Hilliard equation, we establish that the proposed iterative scheme converges with a time step constraint. Further, we numerically demonstrate that the iterative scheme converges for large time steps. The scheme allows for spectral accuracy in space and fast simulation of the dynamics in high dimensions while preserving the discrete form of the energy law. For the general potential well, we present the gradient stable scheme without introducing extra stabilizing terms. Therefore, the numerical error introduced by the operator splitting is reduced to its minimum.