arXiv:2610.29818v1 Announce Type: cross Abstract: The aim of the present study is to develop discontinuous Galerkin and nonconforming finite-element methods for solving the 2D advection-diffusion equation. The discrete formulations are stabilized by means of a family of numerical fluxes and the dispersion relations are constructed via a Fourier/stability analysis. The linear discontinuous P1DG and non conforming P1NC Galerkin discretizations are studied. A long time stability result is proven for the P1NC and P1DG schemes and fluxes examined here by inspecting the discrete frequencies, solutions of the dispersion relations. A super-convergent result is also demonstrated for the P1NC discrete frequency (O(h 4 ) accuracy) compared to the P1DG one (O(h 2 ) accuracy). These results are in line with those obtained in a previous paper for the shallow-water system.