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We are interested in the long-time behaviour of approximate solutions toheterogeneous and anisotropic linear advection-diffusion equations in theframework of hybrid finite volume (HFV) methods on general polygonal/polyhedralmeshes. We consider two linear methods, as well as a new, nonlinear scheme, forwhich we prove the existence and the positivity of discrete solutions. We showthat the discrete solutions to the three schemes converge exponentially fast intime towards the associated discrete steady-states. To illustrate ourtheoretical findings, we present some numerical simulations assessing long-timebehaviour and positivity. We also compare the accuracy of the schemes on somenumerical tests in the stationary case.

Long-time behaviour of hybrid finite volume schemes for advection-diffusion equations: linear and nonlinear approaches