  The Kohn-Sham orbital kinetic energy density $\tau_\sigma(\vec{r}) = \sum_{i}
w_{i\sigma} \big|\nabla \psi_{i\sigma}(\vec{r}) \big|^2$ is one fundamental
quantity for constructing meta-generalized gradient approximations (meta-GGA)
for use by density functional theory. We present a computational scheme of
$\tau_\sigma(\vec{r})$ for full-potential linearized augmented plane wave
method. Our scheme is highly accurate and efficient and easy to implement to
existing computer code. To illustrate its performance, we construct the
Becke-Johnson meta-GGA exchange potentials for Be, Ne, Mg, Ar, Ca, Zn, Kr, Cd
atoms which are in very good agreement with the original results. For bulk
solids, we construct the Tran-Blaha modified Becke-Johnson potential (mBJ) and
confirm its capability to calculate band gaps, with the reported bad
convergence of the mBJ potential being substantially improved. The present
computational scheme of $\tau_\sigma(\vec{r})$ should also be valuable for
developing other meta-GGA's in FLAPW as well as in similar methods utilizing
atom centered basis functions.
