Adjoint-based sensitivity and optimization of flows with shocks

I investigated the primary challenges in constructing consistent, stable, and accurate solutions to adjoint flow equations in the presence of shocks. The inviscid Burgers’ equation was first tested as a relatively simple PDE that exhibits the primary challenges. Results show that an additional artificial viscosity is required to maintain consistency with analytic adjoint solutions. The adjoint one-dimensional Euler equations were also investigated, though no numerical schemes are provably convergent for their adjoint in the presence of discontinuities. A characteristic-based method is proposed, which transforms the adjoint equations into a more-simple set of uncoupled transport equations, which are in principle no different that the adjoint Burgers’ equation. Results show that even highly-dissipative schemes can have inconsistent solutions.