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

Structure preservation in high-order hybrid discretisations of potential-driven advection-diffusion: linear and nonlinear approaches