Sensitivity analysis (SA) is the study of how changes in the input of a model affect the outputs. Standard sensitivity analysis techniques, such as the continuous sensitivity equation (CSE) method cannot be used directly in the framework of hyperbolic PDE systems with discontinuous solution, because it yields Dirac delta functions in the sensitivity solution at the location of state discontinuities. This difficulty is well known from theoretical viewpoint, but only a few works can be found in the literature regarding the possible numerical treatment. Therefore, we investigate in this study how classical numerical schemes for compressible Euler equations can be modified to account for shocks when computing the sensitivity solution. In particular, we propose the introduction of a source term, that allows to remove the spikes associated to the Dirac delta functions in the numerical solution. Numerical studies exhibit a strong impact of the numerical diffusion on the accuracy of this strategy. Therefore, we propose an anti-diffusive numerical scheme coupled with the approximate Riemann solver of Roe for the state problem. For the sensitivity problem, two different numerical schemes are implemented and compared: an HLL-type scheme and an HLLC-type scheme. The effects of the numerical diffusion on the convergence of the schemes with respect to the grid are discussed. Finally, an application of SA to uncertainty quantification is investigated and the different numerical schemes are compared.