  Spatial marching methods, in which the flow state is spatially evolved in the
downstream direction, can be used to produce low-cost models of flows
containing a slowly varying direction, such as mixing layers, jets, and
boundary layers. The parabolized stability equations (PSE) are popular due to
their extremely low cost but can only capture a single instability mode; all
other modes are damped or distorted by regularization methods required to
stabilize the spatial march, precluding PSE from properly capturing non-modal
behavior, acoustics, and interactions between multiple instability mechanisms.
The one-way Navier-Stokes (OWNS) equations properly retain all
downstream-traveling modes at a cost that is a fraction of that of global
methods but still one to two orders of magnitude higher than PSE. In this
paper, we introduce a new variant of OWNS whose cost, both in terms of CPU time
and memory requirements, approaches that of PSE while still properly capturing
the contributions of all downstream-traveling modes. The method is formulated
in terms of a projection operator that eliminates upstream-traveling modes.
Unlike previous OWNS variants, the action of this operator on a vector can be
efficiently approximated using a series of equations that can be solved
recursively, i.e., successively one after the next, rather than as a coupled
set. In addition to highlighting the improved cost scaling of our method, we
derive explicit error expressions and elucidate the relationship with previous
OWNS variants. The properties, efficiency, and accuracy of our method are
demonstrated for both free-shear and wall-bounded flows.
