  This paper deals with the diffusive limit of the Jin and Xin model and its
approximation by an asymptotic preserving finite volume scheme. At the
continuous level, we determine a convergence rate to the diffusive limit by
means of a relative entropy method. Considering a semi-discrete approximation
(discrete in space and continuous in time), we adapt the method to this
semi-discrete framework and establish that the approximated solutions converge
towards the discrete convection-diffusion limit with the same convergence rate.
