  Assumed having axial symmetry, the streamer discharge is often described by a
fluid model in cylindrical coordinate system, which consists of convection
dominated (diffusion) equations with source terms, coupled with a Poisson's
equation. Without additional care for a stricter CFL condition or special
treatment to the negative source term, popular methods used in streamer
discharge simulations, e.g., FEM-FCT, FVM, cannot ensure the positivity of the
particle densities for the cases in attaching gases. By introducing the
positivity-preserving limiter proposed by Zhang and Shu \cite{ppl} and Strang
operator splitting, this paper proposed a finite difference scheme with a
provable positivity-preserving property in cylindrical coordinate system, for
the numerical simulation of streamer discharges in non-attaching and attaching
gases. Numerical examples in non-attaching gas (N$_2$) and attaching gas
(SF$_6$) are given to illustrate the effectiveness of the scheme.
