The subject of this paper is the analysis of the space-time discontinuous Galerkin method for the solution of nonstationary, nonlinear, convection-diffusion problems. In the formulation of the numerical scheme, the nonsymmetric, symmetric and incomplete versions of the discretization of diffusion terms and interior and boundary penalty are used. Then error estimates are briefly characterized. The main attention is paid to the investigation of unconditional stability of the method. An important tool is the concept of the discrete characteristic function. Theoretical results are accompanied by numerical experiments.

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