Efficient linear, Stabilized, second-order time marching schemes for an anisotropic phase field dendritic crystal growth model

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.