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.
Authors
Nicolae Suciu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Surendra Nepal
Department of Mathematics and Computer Science, Karlstad University, Universitetsgatan 2, Karlstad, 65188, Sweden
Yosief Wondmagegne
Department of Mathematics and Computer Science, Karlstad University, Universitetsgatan 2, Karlstad, 65188, Sweden
Magnus Ögren
School of Science and Technology, Örebro University SE-70182, Örebro, Sweden
HMU Research Center, Institute of Emerging Technologies, GR-71004, Heraklion, Greece
Adrian Muntean
Department of Mathematics and Computer Science, Karlstad University, Universitetsgatan 2, Karlstad, 65188, Sweden
Keywords
Global random walk approximation; Stefan-type moving-boundary problems; Finite element approximation, Order of convergence
Paper coordinates
N. Suciu, S. Nepal, Y. Wondmagegne, M. Ögren, A. Muntean, Global random walk for one-dimensional one-phase Stefan-type moving-boundary problems: Simulation results, arXiv:2410.12378, 2024.
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[1] S. C. Gupta, The Classical Stefan Problem: Basic Concepts, Modelling and Analysis, Elsevier 2003. https://doi.org/10.1016/C2017-0-02306-6
[2] J. Stefan, Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere, Ann. Physik Chemie 42 (1891), 269–286. https://doi.org/10.1002/andp.18912780206
[3] D. A Tarzia and C. V. Turner, The one-phase supercooled Stefan problem with a convective boundary condition, Q. Appl. Math. 55(1) (1997), 41–50. https://doi.org/10.1090/qam/1433750
[4] A. Visintin, Models of Phase Transition, Birkhäuser, 1996.
[5] K. Tsunoda, Derivation of Stefan problem from a one-dimensional exclusion process with speed change, Markov Process. Relat. 21 (2015), 263–273.
[6] S. Nepal, Y. Wondmagegne and A. Muntean, Analysis of a fully discrete approximation to a moving-boundary problem describing rubber exposed to diffusants, Appl. Math. Comput. 442 (2023), 127733. https://doi.org/10.1016/j.amc.2022.127733
[7] S. Nepal, R. Meyer, N. H. Kröger, T. Aiki, A. Muntean, Y. Wondmagegne and U. Giese, A moving boundary approach of capturing diffusants penetration into rubber: FEM Approximation and comparison with laboratory measurements, KGK-Kaut. Gumi. Kunst. 5 (2021), 61–69.
[8] D. A. Tarzia, L. T. Villa, On the free boundary problem in the Wen-Langmuir shrinking core model for noncatalytic gas-solid reactions, Meccanica 24 (1989), 86–92. https://doi.org/10.1007/BF01560134
[9] T. Aiki, A. Muntean, Existence and uniqueness of solutions to a matheatical model predicting service life of concrete structures, Advances in Mathematical Sciences and Applications 19(1) (2009), 109–129.
[10] J. D. Evans, J. R. King, The Stefan problem with nonlinear kinetic undercooling, Q. J. Mech. Appl. 56(1) (2003), 139–161. https://doi.org/10.1093/qjmam/56.1.139
[11] S. Nepal, M. Ögren, Y. Wondmagegne and A. Muntean, Random walks and moving boundaries: Estimating the penetration of diffusants into dense rubbers, Probabilist. Eng. Mech. 74 (2023), 103546. https://doi.org/10.1016/j.probengmech.2023.103546
[12] N. Suciu, D. Illiano, A. Prechtel and F.A. Radu, Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media, Adv. Water Resour. 152 (2021), 103935. https://doi.org/10.1016/j.advwatres.2021.103935
[13] N. Suciu, Diffusion in Random Fields. Applications to Transport in Groundwater, Birkhäuser, Cham, 2019. https://doi.org/10.1007/978-3-030-15081-5
[14] N. Suciu and F.A. Radu, Global random walk solvers for reactive transport and biodegradation processes in heterogeneous porous media, Adv. Water Resour. 166 (2022), 104268. https://doi.org/10.1016/j.advwatres.2022.104268
[15] J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, 2004. https://doi.org/10.1137/1.9780898717938
[16] N. Suciu, F.A. Radu and E. Cătinaş, Iterative schemes for coupled flow and transport in porous media – Convergence and truncation errors, Numer. Anal. Approx. Theory 53(1) (2024), 158–183. https://doi.org/10.33993/jnaat531-1429
[17] S. Kutluay, A.R. Bahadir and A. Özdeş, The numerical solution of one-phase classical Stefan problem, J. Comput. Appl. Math. 81(1) (1997), 135–144. https://doi.org/10.1016/S0377-0427(97)00034-4
[18] S. Savović and J. Caldwell, Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions, Int. J. Heat. Mass Tran. 46(15) (2003), 2911–2916. https://doi.org/10.1016/S0017-9310(03)00050-4
[19] M. Mori, A finite element method for solving the two phase Stefan problem in one space dimension, Publ. Res. I. Math. Sci. 13(3) (1977), 723–753. https://doi.org/10.2977/prims/1195189605
[20] M. Mori, Stability and convergence of a finite element method for solving the Stefan problem, Publ. Res. I. Math. Sci. 12(2) (1976), 539–563. https://doi.org/10.2977/prims/1195190728
[21] M.-C. Casabán, R. Company and L. Jódar, Numerical difference solution of moving boundary random Stefan problems, Math. Comput. Simulat. 205 (2023), 878–901. https://doi.org/10.1016/j.matcom.2022.10.026
[22] M. Ögren, Stochastic solutions of Stefan problems with general time-dependent boundary conditions, in: A. Malyarenko, Y. Ni, M. Rančié, S. Silvestrov (Eds.), Stochastic Processes, Statistical Methods, and Engineering Mathematics. SPAS 2019, Springer Proceedings in Mathematics & Statistics, vol. 408, Springer, Cham, 2022. http://dx.doi.org/10.1007/978-3-031-17820-7_29
[23] C.J. Roy, Review of code and solution verification procedures for computational simulation, J. Comput. Phys. 205 (2005), 131–156. https://doi.org/10.1016/j.jcp.2004.10.036
[24] C.D.Alecsa, I. Boros, F. Frank, P. Knabner, M. Nechita, A. Prechtel, A. Rupp and N. Suciu, Numerical benchmark study for flow in heterogeneous aquifers, Adv. Water Resour. 138 (2020), 103558. https://doi.org/10.1016/j.advwatres.2020.103558
[25] P. M. Lewis, Laboratory testing of rubber durability, Polom. Test. 1 (1980), 167–189. https://doi.org/10.1016/0142-9418(80)90002-1
[26] M.J. Hayes and G.S. Park, The diffusion of benzene in rubber. Part 1.—Low concentrations of benzene, T. Faraday Soc. 51 (1955), 1134–1142. https://doi.org/10.1039/TF9555101134
[27] A. V. Kaliyathan, A. V. Rane, S. Jackson and S. Thomas, Analysis of diffusion characteristics for aromatic solvents through carbon black filled natural rubber/butadiene rubber blends, Polym. Composite 42 (2021), 375–396. https://doi.org/10.1002/pc.25832
[28] K. Kumazaki and A. Muntean, Global weak solvability, continuous dependence on data, and large time growth of swelling moving interfaces, Interf. Free Bound. 22(1) (2020), 27–50. https://doi.org/10.4171/ifb/431
[29] S. Nepal, Y. Wondmagegne and A. Muntean, Error estimates for semi-discrete finite element approximations for a moving boundary problem capturing the penetration of diffusants into rubber, Int. J. Numer. Anal. Mod. 19 (2022), 101–125. https://global-sci.org/intro/article_detail/ijnam/20351.html
[30] P. J. Roache, Code verification by the method of manufactured solutions, J. Fluids Eng. 124(1) (2002), 4–10. http://dx.doi.org/10.1115/1.1436090
[31] C. Cuchiero, C. Reisinger and S. Rigger, Implicit and fully discrete approximation of the supercooled Stefan problem in the presence of blow-ups, SIAM J. Numer. Anal. 62(3) (2024), 1145–1170. https://doi.org/10.1137/22M1509722
[32] F. A. Radu, I. S. Pop and S. Attinger, Analysis of an Euler implicit-mixed finite element scheme for reactive solute transport in porous media, Numer. Meth. Part. D. E. 26(2) (2009), 320–344. https://doi.org/10.1002/num.20436