  In this paper we discuss stability properties of various discretizations for
axisymmetric systems including the so called cartoon method which was proposed
by Alcubierre, Brandt et.al. for the simulation of such systems on Cartesian
grids. We show that within the context of the method of lines such
discretizations tend to be unstable unless one takes care in the way individual
singular terms are treated. Examples are given for the linear axisymmetric wave
equation in flat space.
