Continuous and discrete adjoints to the Euler equations for fluids

摘要: Adjoints are used in optimization to speed-up computations, simplify optimality conditions or compute sensitivities. Because time is reversed adjoint equations with first order derivatives, boundary and transmission through shocks can be difficult understand. In this article we analyze the that arise context of compressible flows governed by Euler fluid dynamics. We show continuous adjoints discrete computed automatic differentiation agree numerically; particular found at usually discontinuous contact discontinuities both. Introduction greatly computations; technique has been extensively CFD for optimal control problems shape (see [14, 20, 24, 17, 11] their bibliographies). because convection terms operate opposite direction, Automatic reverse mode [10],[13] automatically generates so problem does not set generator. But then understand make sure there a limit when mesh size tends zero agrees solution. Answering these questions important design supersonic airplanes involved; good case study airplaines least sonic boom ground level. Several investigations have done already B. Mohammadi et al [20], S. Kim [15], A. Loseille [19], M. Nemec [23], Natarajah al.[21, 22] bibliography therein) ∗ Sent publication J. Numer. Methods. Fluid. Mech. †INRIA, 78153 Le Chesnay, France (frederic.alauzet@inria.fr) ‡UPMC-ParisVI, LJLL, 4 Place Jussieu, Paris, F-75005 (pironneau@ann.jussieu.fr)