The advection equation is one of the most important expressions in atmospheric modeling as it appears in many of the primitive equations of the atmosphere such as momentum, continuity and thermodynamic equations ( Mesinger & Arakawa, 1976) and can be written as follows: ∂c/∂t=-u ∂c/∂xv ∂c/∂y where c is the parameter or scalar that is sensed or transported (Engelbrecht, 2000) A Due to the importance of the equation in primitive equations, it is therefore critical that an atmospheric model advection scheme behaves appropriately and it is therefore important to investigate and evaluate how advective processes behave within a model. A non-hydrostatic model is a model in which the hydrostatic assumption, where the vertical component of the pressure gradient force is equal to the gravitational force, is not assumed (Warner, 2011). One such model called the Non-Hydrostatic Sigma Coordinate Model (NSM) was developed in South Africa (Engelbrecht, et al., 2007). The development of a new non-hydrostatic model that explicitly simulates convection is of great importance in South Africa due to the amount of convective activity that occurs in the country. In this model, the advection equations are discretized with the semi-Lagrangian procedure of McGregor (1993) with a time step ∆t (Engelbrecht, et al., 2007). Semi-Lagrangian advection involves representing the real world by both Euler and Lagrangian schemes in order to achieve the smooth resolution of Euler models and the larger time steps of Lagrangian models (Staniforth & Cote, 1991). In this project we will study this new semi-Lagrangian scheme. There are various other advection schemes with different discretization methods. One such scheme involves space-centered, sheet-centered differentiation of the two discretizations of the advection equation with respect to numerical stability using a theoretical analysis of the discretizations. To do this, the harmonic wave substitution method will be used to find a maximum time step that results in stability. Perform the Crowley cone test for both discretizations at various time steps to find the maximum time step that ensures numerical stability. Perform a cone accuracy test in the Crowley cone test for both discretizations in order to determine the conservation of the width and shape of the cone during cone advection by applying root mean square error calculations. Analyze mass conservation during cone advection through the Crowley cone test for both discretizations by calculating the mass of the cone after multiple revolutions of the cone around the origin.