  The Truncated conformal space approach (TCSA) is a numerical technique for
finding finite size spectrum of Hamiltonians in quantum field theory described
as perturbations of conformal field theories. The truncation errors of the
method have been systematically studied near the UV fixed point (when the
characteristic energy related to the coupling is less than the truncation
cutoff) where a good theoretical understanding has been achieved. However
numerically the method demonstrated a good agreement with other methods for
much larger values of the coupling when the RG flow approaches a new fixed
point in the infrared. In the present paper we investigate this regime for a
number of boundary RG flows testing the leading exponent and truncation errors.
We also study the flows beyond the first fixed point which have been observed
numerically but yet lack a theoretical understanding. We show that while in
some models such flows approximate reversed physical RG flows, in other models
the spectrum approaches a stable regime that does not correspond to any local
boundary condition. Furthermore we find that in general the flows beyond the
first fixed point are very sensitive to modifications of the truncation scheme.
