A computational fluid dynamics code relying on a high-order spatial discretization is differentiated using algorithmic differentiation (AD). Two unsteady test cases are considered: a decaying incompressible viscous shear layer and an inviscid compressible flow around a NACA 0012 airfoil. Both tangent and adjoint modes of AD are explored in the viscous case, while only the tangent mode is applied to the inviscid case. The layer of message passing interface (MPI) communications was handled by the AD tool (Tapenade) through the Adjoinable MPI library, with fully automatic inversion of the MPI communications in adjoint mode. A description of the necessary code modifications is provided along with the validation of the computed derivatives and a comparison of the performance of the different codes. The explicit time integration loop of the viscous problem required of the order of 10^6 time steps, which could be inverted in the backward sweep of the adjoint code by means of binomial checkpointing.