  We have studied the phase diagram of the one dimensional $S=1/2$ $XXZ$ model
with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest neighbor
interactions. We have applied the quantum renormalization group (QRG) approach
to get the stable fixed points and the running of coupling constants. The
second order QRG has been implemented to get the self similar Hamiltonian. This
model shows a rich phase diagram which consists of different phases which
possess the quantum spin-fluid and dimer phases in addition to the classical
N\'{e}el and ferromagnetic ones. The border between different phases has been
shown as a projection onto two different planes in the phase space.
