Global random walk for one-dimensional one-phase Stefan-type moving-boundary problems: Simulation results

TL;DR

This paper introduces a global random walk method for simulating one-dimensional Stefan-type moving-boundary problems, demonstrating its effectiveness through benchmark comparisons and applications to material diffusion scenarios.

Contribution

It develops a novel global random walk approach for Stefan problems with kinetic boundary conditions, validated against analytical and finite element solutions.

Findings

Global random walk approximations match benchmark solutions.

The method effectively models diffusion into rubber materials.

Good convergence rates are observed in numerical tests.

Abstract

This work presents global random walk approximations of solutions to one-dimensional Stefan-type moving-boundary problems. We are particularly interested in the case when the moving boundary is driven by an explicit representation of its speed. This situation is usually referred to in the literature as moving-boundary problem with kinetic condition. As a direct application, we propose a numerical scheme to forecast the penetration of small diffusants into a rubber-based material. To check the quality of our results, we compare the numerical results obtained by global random walks either using the analytical solution to selected benchmark cases or relying on finite element approximations with a priori known convergence rates. It turns out that the global random walk concept can be used to produce good quality approximations of the weak solutions to the target class of problems.