  We are interested in the high-order approximation of anisotropic,
potential-driven advection-diffusion models on general polytopal partitions. We
study two hybrid schemes, both built upon the Hybrid High-Order technology. The
first one hinges on exponential fitting and is linear, whereas the second is
nonlinear. The existence of solutions is established for both schemes. Both
schemes are also shown to possess a discrete entropy structure, ensuring that
the long-time behaviour of discrete solutions mimics the PDE one. For the
nonlinear scheme, the positivity of discrete solutions is a built-in feature.
On the contrary, we display numerical evidence indicating that the linear
scheme violates positivity, whatever the order. Finally, we verify numerically
that the nonlinear scheme has optimal order of convergence, expected long-time
behaviour, and that raising the polynomial degree results, also in the
nonlinear case, in an efficiency gain.
