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

Abstract

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. The second one combines the recently developed Invariant Energy Quadratization approach with the linear stabilization technique. Two linear stabilization terms, which are shown to be crucial to remove the oscillations caused by the anisotropic coefficients numerically, are added to enhance the stability while keeping the required accuracy. We further show the obtained linear system is well-posed and prove its unconditional energy stability rigorously. Various 2D and 3D numerical simulations are implemented to demonstrate the stability and accuracy of the schemes.