The stability property of a recently proposed numerical scheme for the Euler-Poisson system P. Crispel, P. Degond and M.-H. Vignal [J. Comput. Phys. 223, No. 1, 208–234 (2007; Zbl 1163.76062)] is analyzed. The linearized Euler-Poisson system is considered about both zero and nonzero velocities and after a Fourier transform in space, the time discretization is applied under scrutiny. The influence of a space-decentered discretization is mimicked by adding viscosity terms in both the mass and momentum conservation equations. It is shown that the resulting scheme is uniformly stable with respect to the Debye parameter. By contrast, other schemes lose the uniform stability property. Elements of a nonlinear stability analysis have been given by considering a model Burgers-Poisson problem.