Finite Volumes for a dissipative free boundary problem
Abstract
We study a toy model for the evolution of the oxygen concentration in an oxide layer. It consists in a transient convection diffusion equation in a one-dimensional domain of variable width. The motions of the boundaries are governed by the traces of the concentration. We exhibit a necessary and sufficient condition on the parameters involved in the model for the existence of a unique traveling-wave solution. Moreover, we show that the model admits some universal entropy structure, in the sense that any convex function of the concentration yields a dissipated free energy (up to exchanges with the outer environment at the boundaries). We propose then an implicit in time arbitrary Lagrangian-Eulerian finite volume scheme based on Scharfetter-Gummel fluxes. It is shown to be unconditionally convergent, to preserve exactly the travelling wave, and to dissipate all the aforementioned free energies. Numerical experiments show that our scheme is first order accurate in time and second order in space, and that the transient solution converges in the long-time limit towards the traveling-wave solution.
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BibTeX
@article{cances2026finite,
title = {{Finite Volumes for a dissipative free boundary problem}},
author = {Clément Cancès and Claire Chainais-Hillairet and Amélie Dupouy},
journal = {Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications},
year = {2026},
month = oct,
url = {https://hal.science/hal-05266770v2},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - JOUR
TI - Finite Volumes for a dissipative free boundary problem
AU - Clément Cancès
AU - Claire Chainais-Hillairet
AU - Amélie Dupouy
PY - 2026
DA - 2026/10/23
JO - Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications
UR - https://hal.science/hal-05266770v2
AB - We study a toy model for the evolution of the oxygen concentration in an oxide layer. It consists in a transient convection diffusion equation in a one-dimensional domain of variable width. The motions of the boundaries are governed by the traces of the concentration. We exhibit a necessary and sufficient condition on the parameters involved in the model for the existence of a unique traveling-wave solution. Moreover, we show that the model admits some universal entropy structure, in the sense that any convex function of the concentration yields a dissipated free energy (up to exchanges with the outer environment at the boundaries). We propose then an implicit in time arbitrary Lagrangian-Eulerian finite volume scheme based on Scharfetter-Gummel fluxes. It is shown to be unconditionally convergent, to preserve exactly the travelling wave, and to dissipate all the aforementioned free energies. Numerical experiments show that our scheme is first order accurate in time and second order in space, and that the transient solution converges in the long-time limit towards the traveling-wave solution.
SN - 1463-9963
SN - 1463-9971
ER -