Continuous and Discrete Adjoints to the Euler Equations for Fluids

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. Introduction In optimization adjoints greatly speed-up computations; the technique has been used extensively in CFD for optimal control problems and shape optimization (see [14, 20, 24, 17, 11] and their bibliographies). Because time is reversed in adjoint equations and because convection terms operate in the opposite direction, boundary conditions and transmission conditions through shocks can be difficult to understand. Automatic differentiation in reverse mode [10],[13] automatically generates the adjoint equations so the problem does not arise because boundary conditions are set by the adjoint generator. But the problem then is to understand and make sure that there is a limit when the mesh size tends to zero and that this limit agrees with the continuous solution. Answering these questions is important for design of supersonic airplanes because shocks are involved; a good case study is the design of airplaines with the least sonic boom at ground level. Several investigations have been done already (see B. Mohammadi et al [20], S. Kim et al [15], A. Loseille et al [19], M. Nemec [23], Natarajah et al.[21, 22] and the bibliography therein) and automatic ∗ Sent for publication to J. Numer. Methods. in Fluid. Mech. †INRIA, 78153 Le Chesnay, France ([email protected]) ‡UPMC-ParisVI, LJLL, 4 Place Jussieu, Paris, F-75005 ([email protected])