Analysis of Discrete Adjoints for Upwind Numerical Schemes

This paper discusses several aspects related to the consistency and stability of the discrete adjoints of upwind numerical schemes. First and third order upwind discretizations of the one-dimensional advection equation are considered in both the finite difference and finite volume formulations. We show that the discrete adjoints may lose consistency and stability near the points where upwinding is changed, and near inflow boundaries where the numerical scheme is changed. The impact of adjoint inconsistency and instability on data assimilation is analyzed.