Fully discrete finite element method with full decoupling structure and second-order temporal accuracy for a flow-coupled dendritic solidification phase-field model

Abstract

The development of effective numerical methods for simulating the dendritic solidification process using the flow-coupled, melt-convective phase-field model has consistently encountered challenges due to the model's complex nonlinear and coupled structures. The major concern of algorithm design is to ensure a numerical scheme that achieves second-order accuracy in time, maintains linearity and attains a fully decoupled structure. All these objectives are accomplished within the framework of guaranteed unconditional energy stability, which is addressed by the fully discrete finite element scheme proposed in this paper. The developed scheme uses the modified projection-type method to deal with the “weighted” Navier-Stokes equations, and the complete decoupling feature is achieved by using the explicit-SAV (Scalar Auxiliary Variable) method, which also helps to linearize the nonlinear potentials. The scheme simplifies its procedure by only requiring the solution of several completely decoupled and linear elliptic equations at every time step, which facilitates its easy implementation. The solvability and energy stability are further rigorously validated. Comprehensive details of the procedural steps for implementation are also provided, accompanied by plenty of numerical tests conducted in both 2D and 3D, serving to numerically ascertain the accuracy and robustness of the scheme.