  We consider numerical approximations for a phase field dendritic crystal
growth model, which is a highly nonlinear system that couples the anisotropic
Allen-Cahn type equation and the heat equation together. We propose two
efficient, linear, second-order time marching schemes. The first one is based
on the linear stabilization approach where all nonlinear terms are treated
explicitly and one only needs to solve two linear and decoupled second-order
equations. Two linear stabilizers are added to enhance the energy stability,
therefore the scheme is quite efficient and stable that allows for large time
steps in computations. The second one combines the recently developed Invariant
Energy Quadratization approach with the linear stabilization approach. Two
linear stabilization terms, which are shown to be crucial to remove the
oscillations caused by the anisotropic coefficients numerically, are added as
well. We further show the obtained linear system is well-posed and prove its
unconditional energy stability rigorously. For both schemes, various 2D and 3D
numerical simulations are implemented to demonstrate the stability and
accuracy.
