Continuous and Discrete Adjoints to the Euler Equations for Fluids
Provides clarity on adjoint behavior at shocks for compressible flow optimization, but is incremental as it validates known theory.
The paper analyzes continuous and discrete adjoint equations for the Euler equations of fluid dynamics, showing they agree numerically and are continuous at shocks but discontinuous at contact discontinuities.
Adjoints are used in optimization to speed-up computations, simplify optimality conditions or compute sensitivities. Because time is reversed in adjoint equations with first order time derivatives, boundary conditions and transmission conditions through shocks can be difficult to understand. In this article we analyze the adjoint equations that arise in the context of compressible flows governed by the Euler equations of fluid dynamics. We show that the continuous adjoints and the discrete adjoints computed by automatic differentiation agree numerically; in particular the adjoint is found to be continuous at the shocks and usually discontinuous at contact discontinuities by both.