Continuous and Discrete Adjoints to the Euler Equations for Fluids

TL;DR

This paper analyzes the properties of continuous and discrete adjoint equations for the Euler equations in fluid dynamics, demonstrating their agreement and behavior at shocks and contact discontinuities.

Contribution

It provides a detailed analysis showing the numerical agreement between continuous and discrete adjoints for Euler equations and clarifies their behavior at discontinuities.

Findings

Continuous and discrete adjoints agree numerically.

Adjoint is continuous at shocks.

Adjoint is usually discontinuous at contact discontinuities.

Abstract

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.