Abstract: We employ Discontinuous Galerkin (DG) methods to solve conservation equations in one and two dimensions on a prototype geodesic grid. DG methods are high order, local, and conservative and have previously been applied on a cubed sphere. Complications arise when moving to an unstructured grid such as the geodesic grid. To address these complications we have developed a 2D transport scheme on arbitrary quadrilateral grids. Each geodesic element can be bi-linearly mapped to a standard reference element. The unstructured grid leads to a more complicated implementation of the Jacobian and flux. In this talk we will overview DG methods and explain the challenges faced when moving from a regular grid to a non-uniform grid. Several sample simulations in one and two dimensions will then be shown.