# Analytic adjoint solutions for the quasi-one-dimensional Euler equations

@article{Giles2001AnalyticAS, title={Analytic adjoint solutions for the quasi-one-dimensional Euler equations}, author={Michael B. Giles and Niles A. Pierce}, journal={Journal of Fluid Mechanics}, year={2001}, volume={426}, pages={327 - 345} }

The analytic properties of adjoint solutions are examined for the quasi-one-dimensional Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is required at the shock. A Green's function approach is used to derive the analytic adjoint solutions corresponding to supersonic, subsonic, isentropic and shocked transonic flows in a converging–diverging…

## 142 Citations

Adjoint characteristics for Eulerian two-dimensional supersonic flow

- MathematicsArXiv
- 2022

The formal expressions, in terms of local flow variables, defining the adjoint characteristic curves, and the associated compatibility relationships satisfied along them, are formally derived in the…

Analytic adjoint solutions for the 2-D incompressible Euler equations using the Green's function approach

- Computer ScienceJournal of Fluid Mechanics
- 2022

The Green's function approach of Giles and Pierce is used to build the lift and drag based analytic adjoint solutions for the two-dimensional incompressible Euler equations around irrotational base flows, and the drag-based adjoint solution turns out to have a very simple closed form in terms of the flow variables.

A Cure for Numerical Instability of Discrete Adjoint Methods to Quasi-1D Flow Equations Near Boundaries

- MathematicsInternational Journal of Aeronautical and Space Sciences
- 2021

A term-by-term comparison has been conducted between the residuals of the continuous and discrete adjoint methods for quasi-1D flow equations. This comparison is done to identify the origin of…

A Note on Adjoint Error Estimation for One – Dimensional Stationary Conservation Laws with Shocks

- Mathematics
- 2012

We consider one-dimensional steady-state conservation laws with discontinuous solutions. Giles and Pierce [7] realized that a shock leads to a new term in the adjoint error representation for target…

Remarks on the numerical solution of the adjoint quasi‐one‐dimensional Euler equations

- Mathematics
- 2012

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…

Analytic Hessian derivation for the quasi-one-dimensional Euler equations

- PhysicsJ. Comput. Phys.
- 2009

Adjoint Formulation for an Embedded-Boundary Cartesian Method

- Computer Science
- 2005

A discrete-adjoint formulation is presented for the three-dimensional Euler equations discretized on a Cartesian mesh with embedded boundaries, to demonstrate the eciency and robustness of the adjoint algorithm for complex-geometry problems.

On the adjoint solution of the quasi‐1D Euler equations: the effect of boundary conditions and the numerical flux function

- Mathematics, Physics
- 2005

This work compares a numerical and analytical adjoint equation method with respect to boundary condition treatments applied to the quasi‐1D Euler equations. The effect of strong and weak boundary…

Adjoint analysis of Buckley-Leverett and two-phase flow equations

- MathematicsComputational Geosciences
- 2018

This paper analyzes the adjoint equations and boundary conditions for porous media flow models, specifically the Buckley-Leverett equation, and the compressible two-phase flow equations in mass…

Ordinary differential equations for the adjoint Euler equations

- MathematicsPhysics of Fluids
- 2022

Ordinary differential equations are derived for the adjoint Euler equations firstly using the method of characteristics in 2D. For this system of partial-differential equations, the characteristic…

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