Shock Equations and Jump Conditions for the 2D Adjoint Euler Equations

This paper considers the formulation of the adjoint problem in two dimensions when there are shocks in the flow solution. For typical cost functions, the adjoint variables are continuous at shocks, wherein they have to obey an internal boundary condition, but their derivatives may be discontinuous. The derivation of the adjoint shock equations is reviewed and detailed predictions for the behavior of the gradients of the adjoint variables at shocks are obtained as jump conditions for the normal adjoint gradients in terms of the tangent gradients. Several numerical computations on a very fine mesh are used to illustrate the behavior of numerical adjoint solutions at shocks.