  Conservation properties of iterative methods applied to implicit finite
volume discretizations of nonlinear conservation laws are analyzed. It is shown
that any consistent multistep or Runge-Kutta method is globally conservative.
Further, it is shown that Newton's method, Krylov subspace methods and
pseudo-time iterations are globally conservative while the Jacobi and
Gauss-Seidel methods are not in general. If pseudo-time iterations using an
explicit Runge-Kutta method are applied to a locally conservative
discretization, then the resulting scheme is also locally conservative.
However, the corresponding numerical flux can be inconsistent with the
conservation law. We prove an extension of the Lax-Wendroff theorem, which
reveals that numerical solutions based on these methods converge to weak
solutions of a modified conservation law where the flux function is multiplied
by a particular constant. This constant depends on the choice of Runge-Kutta
method but is independent of both the conservation law and the discretization.
Consistency is maintained by ensuring that this constant equals unity and a
strategy for achieving this is presented. Experiments show that this strategy
improves the convergence rate of the pseudo-time iterations.
