The subject-matter is the analysis of the discontinuous Galerkin finite element method of lines applied to a linear nonstationary convection–diffusion–reaction problem. In the contrary to the standard FEM the requirement of the conforming properties is omitted. The discretization is carried out with respect to space variables, whereas time remains continuous. In the discontinuous Galerkin discretization, the nonsymmetric stabilization of diffusion terms combined with interior and boundary penalty is applied. In the evaluation of fluxes the idea of upwinding is used. This allows to obtain an optimal error estimate, also verified by numerical experiments.