A phase field method for convective phase change problem preserving maximum bound principle

Numerical simulations of convective solid-liquid phase change problems have long been a complex problem due to the movement of the solid-liquid interface layer, which leads to a free boundary problem. This work develops a convective phase change heat transfer model based on the phase field method. The governing equations consist of the incompressible Navier-Stokes-Boussinesq equations, the heat transfer equation, and the Allen-Cahn equation. The Navier-Stokes equations are penalised for imposing zero velocity within the solid region. For numerical methods, the mini finite element approach (P1b-P1) is used to solve the momentum equation spatially, the temperature and the phase field are approximated by the P1b elements. In the temporal discretization, the phase field and the temperature are decoupled from the momentum equation by using the finite difference method, forming a solvable linear system. A maximum bound principle for the phase field is derived, coming with an estimation of the tolerance of the time step size, which depends on the temperature range. This estimation guides the time step choice in the simulation. The program is developed within the FreeFem++ framework, drawing on our previous work on phase field methods [1] and a mushy-region method toolbox for heat transfer [2]. The accuracy and effectiveness of the proposed method have been validated through real-world cases of melting and solidification with linear or nonlinear buyangcy force, respectively. The simulation results are in agreement with experiments in references.