The coarse grid of numerical weather prediction and climate models requires parametrization models to resolve atmospheric processes that are smaller than the grid size. For parametrization development, these processes are simulated by a high resolution model. At the Royal Netherlands Meteorological Institute, the Dutch Atmospheric Large-Eddy Simulation (DALES) is used. This three-dimensional high resolution model uses advection schemes that are too diffusive when steep gradients are present. In this thesis, an advection scheme based on the Discontinuous Galerkin (DG) method is implemented for DALES. The DG method is known to be dispersive. To remove those non-physical oscillations, the moment limiter of Krivodonova is used. Krivodonova constructed the limiter for one- and two-dimensions. In this thesis the moment limiter and limiting order are derived for three-dimensions. DALES is a model based on the finite difference method and uses operational splitting. Therefore, the DG advection scheme needs a mapping from each cell average to all nodal values that are needed for one DG cell, and a mapping back, which we called mapping a and b respectively. Mappings a that are discussed are taking the cell average as value for all nodal points of the DG cell (cell average a), and taking the L -projection of the cell average to the continuous finite element space (L -projection). This thesis describes mappings b that calculate cell averages of nodal DG values (cell average b) and calculate the cell averages of the tendencies of DG values (cell average of tendency). Using cell average a combined with cell average of tendency, made the DG method as diffusive as the first order upwind scheme. Substituting the cell average a method with the L -projection, the DG method became very dispersive, meaning that there was not enough diffusion. At last, cell average b was tested with the L -projection. Its numerical results showed that the speed of the advection was slower than the theoretical velocity. Therefore, a method is suggested which does not need mappings. An option could be a supergrid that takes multiple DALES cells as a DG cell.