  In this paper, we propose and analyze an efficient numerical method for the
anisotropic phase field dendritic crystal growth model, which is challenging
because we are facing the nonlinear coupling and anisotropic coefficient in the
model. The proposed method is a two-step scheme. In the first step, an
intermediate solution is computed by using BDF schemes of order up to three for
both the phase-field and heat equations. In the second step the intermediate
solution is stabilized by multiplying an auxiliary variable. The key of the
second step is to stabilize the overall scheme while maintaining the
convergence order of the stabilized solution. In order to overcome the
difficulty caused by the gradient-dependent anisotropic coefficient and the
nonlinear terms, some stabilization terms are added to the BDF schemes in the
first step. The second step makes use of a generalized auxiliary variable
approach with relaxation. The Fourier spectral method is applied for the
spatial discretization. Our analysis shows that the proposed scheme is
unconditionally stable and has accuracy in time up to third order. We also
provide a sophisticated implementation showing that the computational
complexity of our schemes is equivalent to solving two linear equations and
some algebraic equations. To the best of our knowledge, this is the cheapest
unconditionally stable schemes reported in the literature. Some numerical
examples are given to verify the efficiency of the proposed method.
