INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2012; 69:966–982 Published online 8 July 2011 in Wiley Online Library (wileyonlinelibrary.com/journal/nmf). DOI: 10.1002/fld.2621 Remarks on the numerical solution of the adjoint quasi-one-dimensional Euler equations C. Lozano * ,† and J. Ponsin Fluid Dynamics Branch, National Institute for Aerospace Technology (INTA), Torrejón de Ardoz, Spain SUMMARY We examine the numerical solution of the adjoint quasi-one-dimensional Euler equations with a central- difference finite volume scheme with Jameson-Schmidt-Turkel (JST) dissipation, for both the continuous and discrete approaches. First, the complete formulations and discretization of the quasi-one-dimensional Euler equations and the continuous adjoint equation and its counterpart, the discrete adjoint equation, are reviewed. The differences between the continuous and discrete boundary conditions are also explored. Second, numerical testing is carried out on a symmetric converging–diverging duct under subsonic flow conditions. This analysis reveals that the discrete adjoint scheme, while being manifestly less accurate than the continuous approach, gives nevertheless more accurate flow sensitivities. Copyright © 2011 John Wiley & Sons, Ltd. Received 4 March 2011; Revised 4 May 2011; Accepted 8 May 2011 KEY WORDS: adjoint equation; design optimization; aerodynamics; differential equations; Euler flow; finite volume; optimization; linear solvers 1. INTRODUCTION For the past 20 years, there has been an increasing interest in the application of adjoint methods in CFD developments. Originally introduced in the CFD arena by Jameson within the context of opti- mal aerodynamic shape design [1–4], this technique has been extended to deal with error analysis and grid adaptation [5–7]. In design applications, the adjoint solution provides the sensitivities of an objective function, such as lift or drag, to a number of design variables that parameterize the shape. These sensitivities can then be used to set up an optimization procedure. In the error control context, the adjoint solution provides the sensitivity of the objective function to errors in the flow discretization. This information can then be used to obtain a posteriori error estimates or to perform adaptive mesh (de)refinement. From a mathematical viewpoint, the (analytic) adjoint equations are obtained from the lineariza- tion of the flow equations. For numerical applications, a discretized version of the adjoint equations is required, which can be formulated in two ways, either by discretizing the analytic adjoint equations (the so-called continuous approach), or by linearizing the discretized flow equations (the discrete approach). Because the operations of linearization and discretization do not commute in general, sensitivity derivatives obtained by using the two approaches may not be identical, with discrete adjoint gradients being consistent with finite-difference gradients independent of the mesh size. On the other hand, the continuous adjoint method has the advantage that the adjoint system has a unique formulation that does not depend on the numerical scheme used to solve the flow equations. *Correspondence to: C. Lozano, Fluid Dynamics Branch, National Institute for Aerospace Technology, Carretera de Ajalvir, Km. 4, 28850 Torrejón de Ardoz, Spain. † E-mail: lozanorc@inta.es Copyright © 2011 John Wiley & Sons, Ltd.