Fully-discrete spectral-Galerkin scheme with decoupled structure and second-order time accuracy for the anisotropic phase-field dendritic crystal growth model

Abstract

In this work, we consider numerical approximations of the anisotropic phase-field dendritic crystal growth model, which is a highly complex coupled nonlinear system consisting of the anisotropic Allen-Cahn equation and the heat equation. Through the combination of a novel explicit auxiliary variable IEQ approach for temporal discretization and the spectral-Galerkin approach for spatial discretization, we develop the first fully-discrete numerical scheme with linearity, decoupled structure, unconditional energy stability, and second-order time accuracy for the particular phase-field dendritic model. In the process of obtaining a full decoupling structure and maintaining energy stability, the introduction of two auxiliary variables and the design of two auxiliary ODEs play a vital role. The designed scheme is highly efficient because only a few elliptic equations with constant coefficients are needed to be solved at each time step. The unconditional energy stability of the scheme has been strictly proved, and the detailed implementation process is given. Through several numerical simulations of 2D and 3D dendritic crystal growth examples, we further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.