  We are interested in the development of an algorithmic differentiation
framework for computing approximations to tangent vectors to scalar and systems
of hyperbolic partial differential equations. The main difficulty of such a
numerical method is the presence of shock waves that are resolved by proposing
a numerical discretization of the calculus introduced in Bressan and Marson
[Rend. Sem. Mat. Univ. Padova, 94:79-94, 1995]. Numerical results are presented
for the one-dimensional Burgers equation and the Euler equations. Using the
essential routines of a state-of-the-art code for computational fluid dynamics
(CFD) as a starting point, three modifications are required to apply the
introduced calculus. First, the CFD code is modified to solve an additional
equation for the shock location. Second, we customize the computation of the
corresponding tangent to the shock location. Finally, the modified method is
enhanced by algorithmic differentiation. Applying the introduced calculus to
problems of the Burgers equation and the Euler equations, it is found that
correct sensitivities can be computed, whereas the application of black-box
algorithmic differentiation fails.
