High-order schemes for cylindrical/spherical geometries with cylindrical/spherical symmetry

In this paper, we implement finite volume Weighted Essentially Non-Oscillatory (WENO) schemes in both cylindrical and spherical coordinate systems for the Euler equations with cylindrical or spherical symmetry. We analyze three different spatial discretizations: one that is shown to be high-order accurate but not conservative, one conservative but not high-order accurate, and one both high-order accurate and conservative. For cylindrical and spherical coordinates, we present convergence results for the advection equation and the Euler equations with an acoustics problem. We then use the Sod shock tube and the Sedov point-blast problems in spherical coordinates coordinates to verify our analysis and implementations. I