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

Nicolae Suciu, Surendra Nepal, Yosief Wondmagegne, Magnus Ögren, Adrian Muntean

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.

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

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.

Paper Structure

This paper contains 14 sections, 24 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. GRW approximation of diffusion
  3. Benchmarking the GRW method for handling moving interfaces
  4. The case of the classical Stefan problem
  5. Problem formulation
  6. GRW convergence study for a benchmark case
  7. The case of the Stefan problem with kinetic condition
  8. Problem formulation
  9. Convergence study for a benchmark case
  10. GRW approximation capturing diffusion in rubber
  11. Problem formulation
  12. Convergence study for a benchmark case for diffusion in rubber
  13. A forecast exercise
  14. Conclusion

Figures (2)

  • Figure 1: Left: Profiles $m(x,t)$ approximated by GRW method. Right: Comparison of the GRW simulation results with the RW and the analytical solutions for the position of the moving boundary $s(t)$.
  • Figure 2: Left: Semilog plot for the profile of $m(x, t)$ as functions of space variable $x$ at time $T = 31$ minutes. Right: Comparison of the position of the moving front approximated by GRW, RW, and FEM with laboratory experimental data.