Marianne Bessemoulin-Chatard

A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme

We propose a finite volume scheme for convection–diffusion equations with nonlinear diffusion. Such equations arise in numerous physical contexts. We will particularly focus on the drift-diffusion system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter–Gummel scheme for nonlinear diffusion. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions.

A FINITE VOLUME SCHEME FOR NONLINEAR DEGENERATE PARABOLIC EQUATIONS

We propose a second order finite volume scheme for nonlinear degenerate parabolic equations which admit an entropy functional. For some of these models (porous media equation, drift-diffusion system for semiconductors) it has been proved that the transient solution converges to a steady state when time goes to infinity. The present scheme preserves steady states and provides a satisfying long time behavior. Moreover, it remains valid and second order accurate in space even in the degenerate case. After describing the numerical scheme, we present several numerical results which confirm the high order accuracy in various regime degenerate and nondegenerate cases and underline the efficiency to preserve the large time asymptotic.

Study of a Finite Volume Scheme for the Drift-Diffusion System. Asymptotic Behavior in the Quasi-Neutral Limit

In this paper, we are interested in the numerical approximation of the classical time-dependent drift-diffusion system near quasi-neutrality. We consider a fully implicit in time and finite volume in space scheme, where the convection-diffusion fluxes are approximated by Scharfetter--Gummel fluxes. We establish that all the a priori estimates needed to prove the convergence of the scheme do not depend on the Debye length

Exponential decay of a finite volume scheme to the thermal equilibrium for drift–diffusion systems

\lambda

Monotone Combined Finite Volume-Finite Element Scheme for a Bone Healing Model

. This proves that the scheme is asymptotic preserving in the quasi-neutral limit

Uniform L ∞ estimates for approximate solutions of the bipolar drift-diffusion system

\lambda \to 0

Preserving monotony of combined edge finite volume-finite element scheme for a bone healing model on general mesh

.

On discrete functional inequalities for some finite volume schemes

Abstract In this paper, we study the large-time behavior of a numerical scheme discretizing drift–diffusion systems for semiconductors. The numerical method is finite volume in space, implicit in time, and the numerical fluxes are a generalization of the classical Scharfetter–Gummel scheme which allows to consider both linear or nonlinear pressure laws. We study the convergence of approximate solutions towards an approximation of the thermal equilibrium state as time tends to infinity, and obtain a decay rate by controlling the discrete relative entropy with the entropy production. This result is proved under assumptions of existence and uniform in time L∞-estimates for numerical solutions, which are then discussed. We conclude by presenting some numerical illustrations of the stated results.

A finite volume scheme for a Keller–Segel model with additional cross-diffusion

We define a combined edge FV-FE scheme for a bone healing model. This choice of discretization allows to take into account anisotropic diffusions and does not impose any restrictions on the mesh. Moreover, following [3], we propose a nonlinear correction to obtain a monotone scheme. We present some numerical experiments which show its good behavior.

NUMERICAL CONVERGENCE RATE FOR A DIFFUSIVE LIMIT OF HYPERBOLIC SYSTEMS: p-SYSTEM WITH DAMPING

We establish uniform \(L^\infty \) bounds for approximate solutions of the drift-diffusion system for electrons and holes in semiconductor devices, computed with the Scharfetter–Gummel finite-volume scheme. The proof is based on a Moser iteration technique adapted to the discrete case.