New efficient time-stepping schemes for the anisotropic phase-field dendritic crystal growth model

TL;DR

This paper introduces new first- and second-order time-stepping schemes for the anisotropic phase-field dendritic crystal growth model, emphasizing efficiency, stability, and decoupling, with detailed analysis and comparison to existing methods.

Contribution

It presents the first totally decoupled, linear, unconditionally stable second-order scheme for this complex crystal growth model.

Findings

Schemes are efficient, requiring only four linear elliptic solves per step.

Proposed methods are unconditionally stable and satisfy a discrete energy law.

The second-order scheme is the first of its kind to be totally decoupled for this model.

Abstract

In this paper, we propose and analyze a first-order and a second-order time-stepping schemes for the anisotropic phase-field dendritic crystal growth model. The proposed schemes are based on an auxiliary variable approach for the Allen-Cahn equation and delicate treatment of the terms coupling the Allen-Cahn equation and temperature equation. The idea of the former is to introduce suitable auxiliary variables to facilitate construction of high order stable schemes for a large class of gradient flows. We propose a new technique to treat the coupling terms involved in the crystal growth model and introduce suitable stabilization terms to result in totally decoupled schemes, which satisfy a discrete energy law without affecting the convergence order. A delicate implementation demonstrates that the proposed schemes can be realized in a very efficient way. That is, it only requires solving four linear elliptic equations and a simple algebraic equation at each time step. A detailed comparison with existing schemes is given, and the advantage of the new schemes are emphasized. As far as we know this is the first second-order scheme that is totally decoupled, linear, unconditionally stable for the dendritic crystal growth model with variable mobility parameter.