On adjoint variables for discontinuous flow

The standard approach to the solution of the adjoint equations stresses the similarity of direct and adjoint equations and implies the use of similar methods for their solution. Nevertheless, the adjoint equations have significant peculiarities in comparison with the direct problem equations at least for compressible flows. From a numerical viewpoint these features concern the existence of the conservative form of the equations, linearity and specific boundary conditions or sources. From the flow field structure viewpoint, there are also sizable differences, for example, the compression shock formation in adjoint variables field is impossible when the rarefaction shock is stable and exists. The latter effect poses some restrictions on the solution of inverse problems.