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

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