Abstract

In this work two simple advection schemes for unstructured meshes are proposed and implemented over spherical icosahedral-hexagonal grids. One of them is fully discrete in space and time while the other one is a semi discrete scheme with third order RungeKutta time integration. Both schemes use cell-wise linear reconstruction. We therefore also present two possible candidates for consistent gradient discretization over general grids. These gradients are designed in a finite volume sense with an adequate modification to guarantee convergence in the absence of a special grid optimization. Monotonicity of the advection schemes is enforced by a slope limiter, at contrast with the widely used of posterior approach of flux-corrected transport (FCT). Convergence of the proposed gradient reconstruction operators is verified numerically. It is found that the proposed modification is indeed necessary for convergence on non-optimized grids. Recently proposed advection test cases are used to evaluate the performance of the slope-limited advection schemes. It is verified that they are convergent and positive. We also compare these schemes to a variant where positivity is enforced by the FCT approach. FCT produces slightly less diffusion but it seems to be at the price of some nonphysical anti-diffusion. These results suggest that the proposed slope limited advection schemes are a viable option for icosahedral-hexagonal grids over sphere.