Third-Order Finite-Difference Schemes on Icosahedral-Type Grids on the Sphere

A practical method is proposed to achieve high-order finite-difference schemes on grids that are quasi-homogeneous on the sphere. A family of grids is used that are characterized by the parameter NP, which can take on values of 3, 4, and 5, etc. The parameter NP is the number of grid patches meeting at the Poles. For NP = 3 the cube sphere grid is obtained and for NP = 5 the icosahedron is obtained. While the grid construction method is valid for all values of NP, the tests performed in this paper concern only the case NP = 5 (i.e., the icosahedron). For each of the rhomboidal patches, the grid is created by connecting points on opposing sides of the rhomboid by great circles. This offers the possibility to obtain derivatives for a line of grid points along a great circle in the classical way. Therefore, it becomes possible to use well-known spatial discretizations from limited-area models. Local models can be transferred to the sphere with rather limited effort. The method was tested using the fourth-order Runge–Kutta integration method and fourth-order spatial differencing. At patch limits, boundary values are obtained using third-order serendipity interpolation, giving the scheme an overall space–time accuracy of 3. The serendipity interpolation is quite efficient. Third-order interpolation in two dimensions is achieved by a set of linear interpolations and a number of function evaluations. All coefficients can be precomputed. The third-order convergence is demonstrated by numerical experiments using Williamson’s test cases 2 and 6.