  Optimal balance is a non-asymptotic numerical method to compute a point on
the slow manifold for certain two-scale dynamical systems. It works by solving
a modified version of the system as a boundary value problem in time, where the
nonlinear terms are adiabatically ramped up from zero to the fully nonlinear
dynamics. A dedicated boundary value solver, however, is often not directly
available. The most natural alternative is a nudging solver, where the problem
is repeatedly solved forward and backward in time and the respective boundary
conditions are restored whenever one of the temporal end points is visited. In
this paper, we show quasi-convergence of this scheme in the sense that the
termination residual of the nudging iteration is as small as the asymptotic
error of the method itself, i.e., under appropriate assumptions exponentially
small. This confirms that optimal balance in its nudging formulation is an
effective algorithm. Further, it shows that the boundary value problem
formulation of optimal balance is well posed up at most a residual error as
small as the asymptotic error of the method itself. The key step in our proof
is a careful two-component Gronwall inequality.
