A note on adjoint error estimation for one-dimensional stationary balance laws with shocks

TL;DR

This paper examines the impact of shocks on adjoint error estimation in one-dimensional steady-state balance laws, showing that the internal boundary condition is naturally satisfied in the vanishing viscosity limit, justifying common computational omissions.

Contribution

It provides a theoretical justification for ignoring the shock-related term in adjoint error estimation by analyzing the vanishing viscosity limit of viscous problems.

Findings

The shock-related term in the adjoint error representation vanishes in the limit.

The internal boundary condition for the adjoint solution is naturally satisfied in the viscous limit.

Justification for ignoring the shock term in practical computations.

Abstract

We consider one-dimensional steady-state balance laws with discontinuous solutions. Giles and Pierce realized that a shock leads to a new term in the adjoint error representation for target functionals.This term disappears if and only if the adjoint solution satisfies an internal boundary condition. Curiously, most computer codes implementing adjoint error estimation ignore the new term in the functional, as well as the internal adjoint boundary condition. The purpose of this note is to justify this omission as follows: if one represents the exact forward and adjoint solutions as vanishing viscosity limits of the corresponding viscous problems, then the internal boundary condition is naturally satisfied in the limit.