  In the context of adjoint-based optimization, nonlinear conservation laws
pose significant problems regarding the existence and uniqueness of both direct
and adjoint solutions, as well as the well-posedness of the problem for
sensitivity analysis and gradient-based optimization algorithms. In this paper
we will analyze the convergence of the adjoint equations to known exact
solutions of the inviscid Burgers' equation for a variety of numerical schemes.
The effect of the non-differentiability of the underlying approximate Riemann
solver, complete vs. incomplete differentiation of the discrete schemes and
inconsistencies in time advancement will be discussed.
