Fully-discrete Spectral-Galerkin numerical scheme with second-order time accuracy and unconditional energy stability for the anisotropic Cahn–Hilliard Model

Abstract

In this work, we construct a fully-discrete Spectral-Galerkin scheme for the anisotropic Cahn–Hilliard model. The scheme is based on the combination of a novel so-called explicit-Invariant Energy Quadratization method for time discretization and the Spectral-Galerkin approach for spatial discretization. The designed scheme only needs to solve several independent linear equations with constant coefficients at each time step, which demonstrates the high computational efficiency. The introduction of two auxiliary variables and the design of their associated auxiliary ODEs play a vital role in obtaining the linear structure and unconditional energy stability and thus avoiding the computation of a variable-coefficient system. The unconditional energy stability of the scheme is further rigorously proved, and the implementation process is given in detail. Through several 2D and 3D numerical simulations, we further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.