On the properties of discrete adjoints of numerical methods for the advection equation

This paper discusses several aspects related to the consistency of the discrete adjoints of upwind numerical schemes. Both linear (finite differences, finite volumes) and nonlinear (slope and flux‐limited) discretizations of the one‐dimensional advection equation are considered. The analysis is focused on uniform meshes and on explicit numerical schemes. We show that the discrete adjoints may lose consistency near the points where upwinding changes, near inflow boundaries where another numerical scheme is employed, and near the locations where the slope/flux limiter is active in the forward simulation. Numerical results are presented to support the theoretical analysis. Copyright © 2007 John Wiley & Sons, Ltd.