  Renormalisation group flows of the bosonic nonlinear \sigma-model are
governed, perturbatively, at different orders of \alpha', by the perturbatively
evaluated \beta--functions. In regions where \frac{\alpha'}{R_c^2} << 1 the
flow equations at various orders in \alpha' can be thought of as \em
approximating the full, non-perturbative RG flow. On the other hand, taking a
different viewpoint, we may consider the abovementioned RG flow equations as
viable {\em geometric} flows in their own right and without any reference to
the RG aspect. Looked at as purely geometric flows where higher order terms
appear, we no longer have the perturbative restrictions . In this paper, we
perform our analysis from both these perspectives using specific target
manifolds such as S^2, H^2, unwarped S^2 x H^2 and simple warped products. We
analyze and solve the higher order RG flow equations within the appropriate
perturbative domains and find the \em corrections arising due to the inclusion
of higher order terms. Such corrections, within the perturbative regime, are
shown to be small and they provide an estimate of the error which arises when
higher orders are ignored.
  We also investigate the higher order geometric flows on the same manifolds
and figure out generic features of geometric evolution, the appearance of
singularities and solitons. The aim, in this context, is to demonstrate the
role of the higher order terms in modifying the flow. One interesting aspect of
our analysis is that, separable solutions of the higher order flow equations
for simple warped spacetimes, correspond to constant curvature Anti-de Sitter
(AdS) spacetime, modulo an overall flow--parameter dependent scale factor. The
functional form of this scale factor (which we obtain) changes on the inclusion
of successive higher order terms in the flow.
