A structure preserving hybrid finite volume scheme for semi-conductor models with magnetic field on general meshes

TL;DR

This paper introduces a novel hybrid finite volume scheme for discretising anisotropic, nonlinear drift-diffusion systems in semiconductor models with magnetic fields, ensuring positivity and exponential convergence on general meshes.

Contribution

A new entropy-based hybrid finite volume scheme that preserves physical bounds and guarantees exponential convergence for anisotropic semiconductor models with magnetic fields.

Findings

Scheme ensures positivity of densities.

Proven exponential convergence to thermal equilibrium.

Numerical tests confirm theoretical properties.

Abstract

We are interested in the discretisation of a drift-diffusion system in the framework of hybrid finite volume (HFV) methods on general polygonal/polyhedral meshes. The system under study is composed of two anisotropic and nonlinear convection-diffusion equations with nonsymmetric tensors, coupled with a Poisson equation and describes in particular semiconductor devices immersed in a magnetic field. We introduce a new scheme based on an entropy-dissipation relation and prove that the scheme admits solutions with values in admissible sets - especially, the computed densities remain positive. Moreover, we show that the discrete solutions to the scheme converge exponentially fast in time towards the associated discrete thermal equilibrium. Several numerical tests confirm our theoretical results. Up to our knowledge, this scheme is the first one able to discretise anisotropic drift-diffusion systems while preserving the bounds on the densities.