A new Lagrangian cell-centered scheme for two-dimensional compressible flows in planar geometry is proposed by Maire et al. The main new feature of the algorithm is that the vertex velocities and the numerical puxes through the cell interfaces are all evaluated in a coherent manner contrary to standard approaches. In this paper the method introduced by Maire et al. is extended for the equations of Lagrangian gas dynamics in cylindrical symmetry. Two different schemes are proposed, whose difference is that one uses volume weighting and the other area weighting in the discretization of the momentum equation. In the both schemes the conservation of total energy is ensured, and the nodal solver is adopted which has the same formulation as that in Cartesian coordinates. The volume weighting scheme preserves the momentum conservation and the area-weighting scheme preserves spherical symmetry. The numerical examples demonstrate our theoretical considerations and the robustness of the new method.