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.

翻译：我们关注于在一般多边形/多面体网格上，采用混合有限体积（HFV）方法对漂移-扩散系统进行离散化。所研究的系统由两个具有非对称张量的各向异性非线性对流-扩散方程构成，并与泊松方程耦合，特别描述了浸没在磁场中的半导体器件。我们提出了一种基于熵-耗散关系的新格式，并证明了该格式的解在允许集内存在——尤其是，计算得到的密度保持正值。此外，我们证明了该格式的离散解在时间上以指数速度收敛于相应的离散热平衡态。多项数值实验验证了我们的理论结果。据我们所知，该格式是首个能够在保持密度有界性的同时，对各向异性漂移-扩散系统进行离散化的格式。