An Upwind-Biased Conservative Transport Scheme for Multistage Temporal Integrations on Spherical Icosahedral Grids

Abstract

standard nominally third-order upwind-biased spatial discretization of the flux-divergence operator was extended to a spherical icosahedral grid. The method can be used with multistage time-stepping schemes such as the Runge?Kutta method to compute the transport of variables on both hexagonal?pentagonal and triangular meshes. Two algorithms can be used to determine mesh cell face values: 1) interpolation using a quadratic function reconstructed subject to an integral constraint, or 2) calculation of the weighted mean of two linearly interpolated and extrapolated values. The first approach was adopted for a triangular mesh because the second approach depends on the mesh having a hexagonal or pentagonal shape. Both approaches were tested on the hexagonal?pentagonal mesh.These schemes were subjected to standard transport tests on a spherical icosahedral grid. A three-stage Runge?Kutta time stepping method was used, and if necessary a flux limiter was applied to maintain monotonicity. The two different methods produced very similar solutions on a hexagonal?pentagonal mesh. Their accuracy was very close to the accuracy of a preexisting method designed for a Voronoi mesh only. When compared to another method that uses a quadratic polynomial interpolation, the phase error of the solutions was reduced, and their accuracy was much improved. The accuracies of the solutions were comparable on triangular and hexagonal?pentagonal meshes.