Add to ORKGWe are interested in the numerical approximation of the bi-temperature Euler equations, which is a non conservative hyperbolic system introduced in [3]. We consider a conservative underlying kinetic model, the Vlasov-BGK-Poisson system. We perform a scaling on this system in order to obtain its hydrodynamic limit. We present a deterministic numerical method to approximate this kinetic system. The method is shown to be Asymptotic-Preserving in the hydrodynamic limit, which means that any stability condition of the method is indepen-dant of any parameter Á, with Á ae 0. We prove that the method is, under appropriate choices, consistant with the solution for bi-temperature Euler. Finally, our method is compared to methods for the fluid model (...