Numerical study of temperature distribution in an inverse moving boundary problem using a meshless method

In this paper, we consider the inverse one-phase one-dimensional Stefan problem to study the thermal processes with phase change in a moving boundary problem and calculate the temperature distribution in the given domain, as well as approximate the temperature and the heat flux on a boundary of the region. For this problem, the location of the moving boundary and temperature distribution on this curve are available as the extra specifications. First, we use the Landau's transformation to get a rectangular domain and then apply the Crank-Nicolson finite-difference scheme to discretize the time dimension and reduce the problem to a linear system of differential equations. Next, we employ the radial basis function collocation technique to approximate the spatial unknown function and its derivatives at each time level. Finally, the linear systems of algebraic equations constructed in this way are solved using the LU factorization method. To show the numerical convergence and stability of the proposed method, we solve two benchmark examples when the boundary data are exact or contaminated with additive noises.

Automated summary

This paper investigates the inverse one-phase one-dimensional Stefan problem to study thermal processes with phase change in moving boundary problems. A numerical method is proposed to calculate temperature distribution and approximate the temperature and heat flux at the boundary. By transforming the domain and discretizing the time dimension using finite-difference schemes, the spatial unknown function is approximated using radial basis function collocation technique, and linear systems of algebraic equations are solved using LU factorization method. The proposed method is validated through numerical convergence and stability analyses on benchmark examples with exact or noisy boundary data.