Summary: Continuous and Discrete Adjoints to the Euler
Equations for Fluids
April 8, 2011
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.
In optimization adjoints greatly speed-up computations; the technique has been
used extensively in CFD for optimal control problems and shape optimization
(see [12, 14, 15, 20, 10] and their bibliographies). Because time is reversed in