  Previous papers have shown the impact of partial convergence of discretized
PDE on the accuracy of tangent and adjoint linearizations. A series of papers
suggested linearization of the fixed point iteration used in the solution
process as a means of computing the sensitivities rather than linearizing the
discretized PDE, as the lack of convergence of the nonlinear problem indicates
that the discretized form of the governing equations has not been satisfied.
These works showed that the accuracy of an approximate linearization depends in
part on the convergence of the nonlinear system. This work shows an error
analysis of the impact of the approximate linearization and the convergence of
the nonlinear problem for both the tangent and adjoint modes and provides a
series of results for an exact Newton solver, an inexact Newton solver, and a
low storage explicit Runge-Kutta scheme to confirm the error analyses.
