This study is focused on a Fourier stability and accuracy analysis of the time integration algorithms using generalized trapezioidal family of methods of scalar unsteady convection–diffusion equation with periodic boundary conditions. The discretization in space dimension is performed by standard Galerkin finite element formulation for low Peclet numbers and stabilized finite element formulation for large Peclet numbers. The stability analysis is performed namely by von-Neumann stability analysis. Accuracy is measured in terms of damping errors and phase speed errors. The behaviour of these temporal errors of the particular time stepping algorithms, i.e. forward Euler, Crank-Nicolson and backward Euler methods are compared with each other. Particular attention is given to the stabilized finite element formulation, that is the case where we consider high Peclet numbers. For this case, it is concluded that the Crank-Nicolson time stepping represents a better approximate solution compared to the other time integrators on transport process of an initial wave profile. Finally, at the end of the study, we derive a stabilization parameter under a particular condition on Courant number, which provides the relative phase speed error being almost equivalent to its optimal level, that is, the waves with different Fourier modes propagate almost in the same speed. Theoretical results are confirmed by a number of numerical experiments.