Interesting result on discrete adjoints: They show that discrete adjoint for scalar conservation law converges pointwise everywhere except at shocks, provided the numerical dissipation increases as the grid is refined. The number of points across the shock must increase as the grid is refined.

http://link.aip.org/link/?SNA/48/882/1&agg=rss

Convergence of Linearized and Adjoint Approximations for Discontinuous Solutions of Conservation Laws. Part 1: Linearized Approximations and Linearized Output Functionals

SIAM Journal on Numerical Analysis on 6/29/10
Mike Giles and Stefan Ulbrich

This paper analyzes the convergence of discrete approximations to the linearized equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. A simple modified LaxFriedrichs discretization is used on a uniform grid, and a key point is that the numerical smooth ... [SIAM J. Numer. Anal. 48, 882 (2010)] published Tue Jun 29, 2010.

-------------------------

http://link.aip.org/link/?SNA/48/905/1&agg=rss

Convergence of Linearized and Adjoint Approximations for Discontinuous Solutions of Conservation Laws. Part 2: Adjoint Approximations and Extensions

Mike Giles and Stefan Ulbrich

This paper continues the convergence analysis in [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882904] of discrete approximations to the linearized and adjoint equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. We consider a simple m ... [SIAM J. Numer. Anal. 48, 905