Abstract

This paper studies a mathematical model of the phase transition process as a moving boundary problem (MBP). An equation describing the temperature-dependent thermal conductivity and Robin-type boundary condition at the fixed boundary is used in the model. In order to solve the considered MBP, we propose an iterative-based Keller box method (KBM) for the numerical approximation and boundary immobilization method (BIM) to immobilize the moving boundary. In addition, KBM incorporates nonlinearities in thermal conductivity and boundary conditions. We study the stability and consistency and found out that the scheme is second-order accurate in both time and space. A specific case having a similarity solution has been considered to validate the numerical scheme. Our numerical results show that the KBM is in good agreement with the similarity solution. In addition, the rate of convergence of the KBM scheme is two. Different parameters and temperature-dependent thermal conductivity are studied to determine how these affect the position of the moving boundary.

中文翻译：

---

具有对流边界条件的非线性移动边界问题的迭代数值方法

摘要

本文研究了作为移动边界问题（MBP）的相变过程的数学模型。模型中使用了描述与温度相关的热导率和固定边界处的 Robin 型边界条件的方程。为了解决所考虑的MBP，我们提出了一种基于迭代的凯勒盒法（KBM）用于数值近似和边界固定法（BIM）来固定移动边界。此外，KBM 还考虑了热导率和边界条件的非线性。我们研究了稳定性和一致性，发现该方案在时间和空间上都是二阶精度的。考虑了具有相似解的特定情况来验证数值方案。我们的数值结果表明 KBM 与相似解非常一致。另外，KBM方案的收敛率为2。研究不同的参数和与温度相关的热导率，以确定它们如何影响移动边界的位置。