  Conservation and consistency are fundamental properties of discretizations of
systems of hyperbolic conservation laws. Here, these concepts are extended to
the realm of iterative methods by formally defining locally conservative and
flux consistent iterations. These concepts are of both theoretical and
practical importance: Based on recent work by the authors, it is shown that
pseudo-time iterations using explicit Runge-Kutta methods are locally
conservative but not necessarily flux consistent. An extension of the
Lax-Wendroff theorem is presented, revealing convergence towards weak solutions
of a temporally retarded system of conservation laws. Each equation is modified
in the same way, namely by a particular scalar factor multiplying the spatial
flux terms. A technique for enforcing flux consistency, and thereby recovering
convergence, is presented. Further, local conservation is established for all
Krylov subspace methods, with and without restarts, and for Newton's method
under certain assumptions on the discretization. Thus it is shown that
Newton-Krylov methods are locally conservative, although not necessarily flux
consistent. Numerical experiments with the 2D compressible Euler equations
corroborate the theoretical results. Further numerical investigations of the
impact of flux consistency on Newton-Krylov methods indicate that its effect is
case dependent, and diminishes as the number of iterations grow.
