Technical notes and discussions on implementing and testing A-grid and C-grid icosahedral shallow water model solvers on sphere

The lack of detailed technical notes, discussions and computer software in the public domain for icosahedral grid generators and the analytical formula of the relatively complex Williamson test cases has posed inconvenience for the community to verify and systematically improve numerical algorithms for atmospheric dynamics on icosahedral grids. This work intends to bridge the gap. We discuss technical details encountered when implementing accurate A-grid and C-grid shallow-water model solvers on icosahedral grids on sphere. Our major contributions include (i) adding centripetal force to the spring dynamics grid generator to reduce the global angle variance for spherical polygons, which significantly enhances model stability, (ii) introducing a new type of gravity center for spherical polygons that reduces the finite volume error for the gravity centered grid in the A-grid staggering approach, (iii) providing the explicit analytical formula for the shallow-water model test case 4 (the forced nonlinear system with a translating low), (iv) illustrating a self-contained computer code that implements the icosahedral A-grid and C-grid shallow-water model solvers within a single framework, including grid generators, shallow-water model test case initial conditions, time integration schemes, a linear order spherical interpolation method, and error analysis modules.