Entropy solutions of a diffusion equation with discontinuous hysteresis and their finite volume approximation
Solutions entropiques d'une équation de diffusion avec hystérésis discontinue et leur approximation par volumes finis
Résumé
We provide a finite volume approximation in dimension $d \geq 1$ of a problem involving a quasilinear parabolic equation with discontinuous hysteresis modeling a phase change. The model is related to a singular limit of the pseudo-parabolic regularization of an ill-posed forward-backward diffusion equation. Besides, an important feature of the model lies in the (possibly moving) interface triggered by an initial separation of phases. The convergence of the numerical solution to a suitable weak entropy solution of the forward-backward problem is shown under a parallelism assumption between the nonlinearities driving the (diffusive) evolution in each phase. The main challenge lies in the treatment of the discontinuous hysteresis operator in the proof of the compactness of the sequence of numerical solutions. This is achieved by regularizing the hysteresis operator with a continuous one for which Hilpert inequalities are accessible and allow us to obtain crucial uniform translation estimates in $L^1$ in space. Numerical simulations, computed using a Julia-based framework developed for the finite volume discretization of reaction-diffusion equations, are provided.
| Origine | Fichiers produits par l'(les) auteur(s) |
|---|---|
| Licence |