Report on GPU solver for Fourier Pseudo-spectral method on Cahn-Hilliard equation

Numerical method for Cahn-Hilliard equation has been well-studied, but few can be generalized to fractional Cahn-Hilliard equation. In this project to modified the numerical method proposed by Brian Wetton et al in the paper High accuracy solutions to energy gradient flows from material science models[1]. The method they described in the paper is a pseudo-spectral method suitable for considering fractional laplacian. Mathematically speaking, it is a implicit time stepping method using adaptive time step. At each time step, using conjugate gradient to solve the minimization problem. However, in typical interesting cases like Cahn-Hilliard and Fractional Cahn-Hilliard equation, PDE contains non-linear term, which is not suitable for Fourier method in general. A way to get around this is linearize the non-linear term, use the modified discretization as a pre-conditioner for the conjugate gradient step. The discretization is as follow: U − km[ ∆h∆hU + ∆h W ′(Um) ]− Um−1 = 0 (3)