  $3+1$ formulations of the Einstein field equations have become an invaluable
tool in Numerical relativity, having been used successfully in modeling
spacetimes of black hole collisions, stellar collapse and other complex
systems. It is plausible that similar considerations could prove fruitful for
modified gravity theories. In this article, we pursue from a numerical
relativistic viewpoint the $3+1$ formulation of metric $f(R)$ gravity as it
arises from the fourth order equations of motion, without invoking the
dynamical equivalence with Brans-Dicke theories. We present the resulting
system of evolution and constraint equations for a generic function $f(R)$,
subject to the usual viability conditions. We confirm that the time propagation
of the $f(R)$ Hamiltonian and Momentum constraints take the same Mathematical
form as in general relativity, irrespective of the $f(R)$ model. We further
recast the 3+1 system in a form akin to the BSSNOK formulation of numerical
relativity. Without assuming any specific model, we show that the ADM version
of $f(R)$ is weakly hyperbolic and is plagued by similar zero speed modes as in
the general relativity case. On the other hand the BSSNOK version is strongly
hyperbolic and hence a promising formulation for numerical simulations in
metric $f(R)$ theories.
