In this paper, we studied discontinuous Galerkin method as well as its application of linear evolutionary convection diffusion equation. We shown error estimate results under different variances of equation and boundary conditions. Furthermore, it was stated that there exists uniform error estimate with respect to the diffusion coefficient \(\epsilon\rightarrow 0_+\) for linear convection diffusion reaction equation. We introduced SUPG method used for stabilization of discontinuous elements for large Péclet numbers. We presented two code snippets which implement two crucial steps when solving equations using DUNE framework: grid definition, which is used for representing discretization of a domain, and instantiation of objects needed for numerization process. Within most models describing some physical phenomena, convective part is dominant compared to diffusion part. It leads to a well posed continuous problem which, according to Lax-Milgram lemma, has a unique solution. Numerical solutions obtained using conforming method of finite elements, however, suffered from severe instabilities. They often contain oscillations which only amplify through time. Quality of discontinuous Galerkin method was confirmed by numerical experiment for large Péclet numbers, contrary to finite element method which yields boundary layers in solution and requires extra sophisticated stabilization techniques.