An implicit regularized enthalpy Lattice Boltzmann Method for the Stefan problem

TL;DR

This work tackles the classical Stefan problem with phase change by embedding a regularized enthalpy formulation into the Lattice Boltzmann Method. The proposed implicit regularized enthalpy based method (IREBM) introduces a smooth liquid-fraction function and a local Newton solver to handle the nonlinear source term, preserving locality and enabling scalable computation. Chapman–Enskog analysis and Taylor expansions confirm that IREBM recovers the intended macroscopic equation with a regularized latent term, while numerical tests in 1D and 2D show reduced interface oscillations and competitive accuracy relative to existing enthalpy-based LB schemes. The approach achieves accuracy and stability improvements with a modest computational overhead, and the authors highlight its potential for 3D and GPU implementations in future work.

Abstract

Solving the Stefan problem, also referred as the heat conduction problem with phase change, is a necessary step to solve phase change problems with convection. In this article, we are interested in using the Lattice Boltzmann Method (LBM) to solve the Stefan problem using a regularized total enthalpy model. The liquid fraction is treated as a nonlinear source/sink term, that involves the time derivative of the solution. The resulting non-linear system is solved using a Newton algorithm. By conserving the locality of the problem, this method is highly scalable, while keeping a high accuracy. The newly developed scheme is analyzed theoretically through a Chapman-Enskog expansion and illustrated numerically with 1D and 2D benchmarks.

Figures (11)

• Figure 1: Sketch of the simulation configuration.
• Figure 2: Initial solution for the 1D case.
• Figure 3: Position of the interface (left) and error with respect to the exact solution (right). Comparison between EEBM, ILFBM, IREBM (with $\delta=0.005$) $N = 801$.
• Figure 4: Position of the interface (left) and error with respect to the exact solution (right). Comparison of IREBM for three values of $\delta$ ($\delta_\text{min}=0.005$), for $N = 801$.
• Figure 5: Position of the interface (left) and error with respect to the exact solution (right). Comparison of different mesh resolutions for IREBM, with $\delta=0.005$.