The sensitivity of a model variable to a given parameter can be defined as the partial derivative of this variable with respect to the parameter. The classical empirical (or finite difference) approach to assess the sensitivity, consists in performing two simulations with two slightly different values of the parameter of interest and computing the difference between the results, normalized by the parameter variation. This method, however, has three main drawbacks: (i) two simulations are needed to compute the sensitivity, (ii) numerical artefacts may be present in the sensitivity solution, and (iii) the flow variables must be assumed continuous and differentiable with respect to the parameter. Such an assumption is wrong in presence of hydraulic jumps or phenomena such as tidal bores, yielding infinite sensitivity values when applying the empirical method. A direct, finite volume-based approach is proposed to overcome these problems, with a modified HLLC solver adapted to the resolution of sensitivity equations (HLLS), including a specific treatment at discontinuities locations in the flow solution. It is applied to the 2D Shallow Water Equations (SWEs) and to the 1D SWEs with discontinuities. Numerical applications (tidal bore propagation, dam-break, 2D-simulations...) demonstrate the efficiency of the proposed solver and a good agreement between numerical and analytical solutions, when they exist. Besides, some artefacts of the empirical solution are eliminated by the direct method.