Riemann invariants of adjoint equations of shallow water equations
 

hi,friends. recently i use discontinuous galerkin method to compute a control problem of 2D shallow water equations. when i compute the adjoint equations i find that the boundary conditions of adjoint equation is difficult to deal with. we know DG apply Riemann problem to compute numerical flux which need two states of any side. as for the primal equations we use Riemann invariants to compute those unkown varibles on the boundary of computing area such as inlet and outlet boundary conditions. as for the adjoint equations, however, i don't know what those Riemann invariants are on the inlet and outlet boundary?
Thank you for your attention!