An Eulerian flux-form advection scheme, called the Two-step Shape-Preserving Advection Scheme(TSPAS), was generalized and implemented on a spherical icosahedral hexagonal grid(also referred to as a geodesic grid) to solve the transport equation. The C grid discretization was used for the spatial discretization. To implement TSPAS on an unstructured grid,the original finite-difference scheme was further generalized. The two-step integration utilizes a combination of two separate schemes(a low-order monotone scheme and a high-order scheme that typically cannot ensure monotonicity) to calculate the fluxes at the cell walls(one scheme corresponds to one cell wall). The choice between these two schemes for each edge depends on a pre-updated scalar value using slightly increased fluxes. After the determination of an appropriate scheme, the final integration at a target cell is achieved by summing the fluxes that are computed by the different schemes. The conservative and shape-preserving properties of the generalized scheme are demonstrated. Numerical experiments are conducted at several horizontal resolutions. TSPAS is compared with the Flux Corrected Transport(FCT) approach to demonstrate the differences between the two methods, and several transport tests are performed to examine the accuracy, efficiency and robustness of the two schemes.
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