Citation: Ancourt, K.; Peter, J.; Atinault, O. Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows. Aerospace 2023, 10, 797. https://doi.org/10.3390/ aerospace10090797 Academic Editor: Carlos Lozano Received: 20 July 2023 Revised: 8 September 2023 Accepted: 10 September 2023 Published: 12 September 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). aerospace Article Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows Kevin Ancourt 1 , Jacques Peter 1, * and Olivier Atinault 2 1 Département Aérodynamique, Aéroélasticité et Acoustique (DAAA), ONERA, Université Paris Saclay, 29 Avenue de la Division Leclerc, Châtillon, 92322 Paris, France; kevin.ancourt@onera.fr 2 Département Aérodynamique, Aéroélasticité et Acoustique (DAAA), ONERA, Université Paris Saclay, 8 Rue des Vertugadins, 92190 Meudon, France; olivier.atinault@onera.fr * Correspondence: jacques.peter@onera.fr Abstract: The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to present a linear algebra analysis that greatly simplifies the discussion of the number of independent characteristic equations satisfied along a family of characteristic curves. This method may be applied for both the direct and the adjoint problem. Our second goal is to directly derive in conservative variables the characteristic equations of 2D compressible inviscid flows. Finally, the theoretical results are assessed for a nozzle flow with a classical scheme and its dual consistent discrete adjoint. Keywords: continuous adjoint; inviscid flow; compressible flow; method of characteristics; characteristic equation; characteristic curve 1. Introduction The method of characteristics is a well-known method for studying partial differential equations (PDEs). It aims to exhibit specific hypersurfaces in the input domain where the solution of the PDE of interest satisfies an ordinary differential equation (ODE). When applied to 2D steady-state inviscid compressible flows, it is known to provide a full resolution of a supersonic area based only on the knowledge of the inflow (whereas it provides partial information for a subsonic flow) [1–4]. Both a theoretical understanding and practical calculations of a variety of flows (in nozzles, along steps, along curved walls) are enabled by this technique. In addition, discrete and continuous adjoint are now well-established methods for shape optimization [5–9] and goal-oriented mesh adaptation [10–12]. Because of these important applications, regular efforts have been devoted to the fast and safe writing of adjoint modules [13–16] and to the efficient solving of the adjoint equations [17,18]. Adjoint methods are also useful for flow control [19,20], meta-modeling [21,22], receptivity– sensitivity–stability analyses [23,24], and data assimilation [25,26]. Moving from direct to adjoint Euler equations, the classical flux Jacobians are replaced in the PDE by the opposite of their transpose. This allows the simple adaptation in 2D and 3D of the characteristic equations in the specific sense of the quasi-one-dimensional scalar propagation equations satisfied in the direction normal to a boundary [27]. From the earliest works on the continuous adjoint method, it has been proved that for the adjoint problem (a) information travels along characteristics in the opposite direction to the flow, and (b) the right eigenvectors of the local Jacobian replace the left eigenvectors in the equation definition [5,28,29]. In addition, in a recent paper, the characteristic equations (CEs), in the sense of nonlinear ODEs satisfied along curves to be defined, have been derived for the adjoint 2D steady-state Euler equations [30]. Aerospace 2023, 10, 797. https://doi.org/10.3390/aerospace10090797 https://www.mdpi.com/journal/aerospace