  We carry out a stability analysis for the real space split operator method
for the propagation of the time-dependent Klein-Gordon equation that has been
proposed Ruf et al. [M. Ruf, H. Bauke, C.H. Keitel, A real space split operator
method for the Klein-Gordon equation, Journal of Computational Physics 228 (24)
(2009) 9092-9106, doi:10.1016/j.jcp.2009.09.012]. The region of algebraic
stability is determined analytically by means of a von-Neumann stability
analysis for systems with homogeneous scalar and vector potentials. Algebraic
stability implies convergence ofthe real space split operator method for smooth
absolutely integrable initial conditions. In the limit of small spatial grid
spacings $h$ in each of the $d$ spatial dimensions and small temporal steps
$\tau$, the stability condition becomes $h/\tau>\sqrt{d}c$ for second order
finite differences and $\sqrt{3}h/(2\tau)>\sqrt{d}c$ for fourth order finite
differences, respectively, with $c$ denoting the speed of light. Furthermore,
we demonstrate numerically that the stability region for systems with
inhomogeneous potentials coincides almost with the region of algebraic
stability for homogeneous potentials.
