Efficient numerical scheme with full decoupling, second-order temporal accuracy and unconditional energy stability for a flow-coupled melt-convective dendritic solidification phase-field model

Developing efficient numerical methods for the flow-coupled, melt-convective phase-field dendritic solidification model has consistently been a challenging 问题, attributed to its complex, highly coupled nonlinear nature. The central issue in algorithm design revolves around the time-marching aspect, specifically, the development of a robust time-marching method that ensures second-order time accuracy, unconditional energy stability, linearity, and full decoupling. We achieve such a scheme by integrating a modified projection approach of the Navier-Stokes equations and the explicit-IEQ (invariant energy quadratization) method of nonlinear potentials. At each time step, we only need to solve a series of fully decoupled linear elliptic equations, so the obtained algorithm is very easy to implement. We rigorously establish the solvability and unconditional energy stability of the developed scheme. Detailed descriptions of the implementation process are provided, alongside extensive numerical simulations in both 2 dimensions and 3 dimensions, to numerically validate the scheme's accuracy and stability.