  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.
