  We present an accurate and efficient formulation for the calculation of
phonons in real-space Kohn-Sham density functional theory. Specifically,
employing a local exchange-correlation functional, norm-conserving
pseudopotential in the Kleinman-Bylander representation, and local form for the
electrostatics, we derive expressions for the dynamical matrix and associated
Sternheimer equation that are particularly amenable to the real-space
finite-difference method, within the framework of density functional
perturbation theory. In particular, the formulation is applicable to insulating
as well as metallic systems of any dimensionality, enabling the efficient and
accurate treatment of semi-infinite and bulk systems alike, for both orthogonal
and non-orthogonal cells. We also develop an implementation of the proposed
formulation within the high-order finite-difference method. Through
representative examples, we verify the accuracy of the computed phonon
dispersion curves and density of states, demonstrating excellent agreement with
established planewave results.
