Quantum mechanics

Anisotropy

Phase (matter)

Scheme (mathematics)

Pure mathematics

Mathematical physics

Materials science

Applied mathematics

Field (mathematics)

Projection method

Mathematical optimization

Thermodynamics

Physics

Nonlinear system

Mathematics

Invariant (physics)

Numerical stability

Dykstra's projection algorithm

Phase field models

Mathematical analysis

Numerical analysis

Mechanics

Convection

Dissipative system

Journal of Computational Physics

In this paper, we consider numerical approximations for a dendritic solidification phase field model with melt convection in the liquid phase, which is a highly nonlinear system that couples the anisotropic Allen-Cahn type equation, the heat equation, and the weighted Navier-Stokes equations together. We first reformulate the model into a form which is suitable for numerical approximations and establish the energy dissipative law. Then, we develop a linear, decoupled, and unconditionally energy stable numerical scheme by combining the modified projection scheme for the Navier-Stokes equations, the Invariant Energy Quadratization approach for the nonlinear anisotropic potential, and some subtle explicit-implicit treatments for nonlinear coupling terms. Stability analysis and various numerical simulations are presented.

Efficient numerical scheme for a dendritic solidification phase field model with melt convection