A stability analysis of a real space split operator method for the Klein-Gordon equation

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.