S. Polavarapu et M. Tanguay, LINEARIZING ITERATIVE PROCESSES FOR 4-DIMENSIONAL DATA-ASSIMILATION SCHEMES, Quarterly Journal of the Royal Meteorological Society, 124(549), 1998, pp. 1715-1742
The tangent linear model is used in applications including Kalman filt
ers and the growth of perturbations. It is also used to define adjoint
models in applications such as four-dimensional variational assimilat
ion or sensitivity studies. The validity of the tangent linear model f
or all of these applications is determined by the period of time and i
nitial amplitudes for which perturbation growth remains linear. In thi
s work we examine the validity of various linearizations of a class of
iterations known as fixed-point iterations. The tangent Linearization
is found to converge more slowly than the nonlinear iteration, and al
so requires all iterates of the nonlinear process. The tangent lineari
zation can also be invalid if too few iterations of the nonlinear proc
ess are taken. An approximate linearization which requires only the la
st iterate of the nonlinear process was found to be accurate when the
correct Linearization has converged. Application of these results to i
terative processes occurring in a numerical weather-prediction model r
eveals that the approximate linearization can be effective. Specifical
ly, the linearization of the iteration for the mid-trajectory position
for semi-Lagrangian advection was found to converge more slowly than
the nonlinear iteration, for our choice of initial guess. For the iter
ative solution of an elliptic equation, the maximum number of iteratio
ns allowed by the model was sufficient for both the nonlinear and line
arized iterations. Thus the approximate linearization could be used to
save CPU time or memory space.