An accurate and parallel method with post-processing boundedness control for solving the anisotropic phase-field dendritic crystal growth model

Abstract

A fast, accurate, and stable numerical algorithm is proposed to solve the anisotropic phase-field dendritic crystal growth model. The algorithm combines the first-order direction splitting method and the linear stabilization technique, which preserve the energy stability while ensuring fast computing. The related stability analysis is shown. To obtain high accurate solution, the solution of first-order direction splitting method is extrapolated to be fourth-order, and the fourth-order difference method is applied to spatial discretization. To further improve the stability and prevent the blow-up of numerical solution, two post-processing methods, the bound preserving least-distance modification and the cut-off approach, are developed to control the boundedness of numerical solution, and their theoretical proofs are given. The proposed novel algorithm can be performed in parallel in each time step, significantly improving computing efficiency. Several numerical examples demonstrate the effectiveness of the proposed algorithm, including convergence and stability tests, the effectiveness of two post-processing techniques, and a series of 2D and 3D dendritic crystal growth simulations.