Stochastic solutions of Stefan problems

TL;DR

This paper introduces a stochastic random walk approach to solving one-dimensional Stefan problems with time-dependent boundary conditions, demonstrating statistical convergence and potential advantages in higher-dimensional applications.

Contribution

The novel aspect is modeling the moving boundary in Stefan problems using a stochastic random walk method, offering a flexible and easily parallelizable alternative to traditional techniques.

Findings

Statistical convergence observed as x  0

Method is competitive for large domains and higher dimensions

Approach is adaptable to various applications like climate and battery modeling

Abstract

This work deals with the one-dimensional Stefan problem with a general time-dependent boundary condition at the fixed boundary. Stochastic solutions are obtained using discrete random walks, and the results are compared with analytic formulae when they exist, otherwise with numerical solutions from a finite difference method. The innovative part is to model the moving boundary with a random walk method. The results show statistical convergence for many random walkers when $\Delta x \rightarrow 0$. Stochastic methods are very competitive in large domains in higher dimensions and has the advantages of generality and ease of implementation. The stochastic method suffers from that longer execution times are required for increased accuracy. Since the code is easily adapted for parallel computing, it is possible to speed up the calculations. Regarding applications for Stefan problems, they have historically been used to model the dynamics of melting ice, and we give such an example here where the fixed boundary condition follows data from observed day temperatures at \"{O}rebro airport. Nowadays, there are a large range of examples of applications, such as climate models, the diffusion of lithium-ions in lithium-ion batteries and modelling steam chambers for petroleum extraction.