Numerical Methods

Consider finite volume scheme on unstructured grids for the Euler equations.

Let be normal to a cell face and whose magnitude is equal to face area. Let be the conserved vector. The finite volume update equation using forward Euler time discretization is



Here is a normal vector pointing from current cell "j" into the neighbouring cell...

Adjoint Consistency Analysis of Discontinuous Galerkin Discretizations
SIAM J. Numer. Anal. Volume 45, Issue 6, pp. 2671-2696 (2007)
http://dx.doi.org/10.1137/060665117

This paper gives a notion of consistent adjoint discretizations for DG methods. However the definition seems to be restricted to smooth solutions only. For the case of a conservation law, they show that boundary conditions of the primal problem must be appropriate for the corresponding adjoint discretization...

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...