On the use of reduced grids in conjunction with the Equator-Pole grid system

This is a continuation of a previous paper by Starius, where for the solution of the shallow water equations on the sphere, we consider Equator-Pole grid systems consisting of one latitude-longitude grid covering an annular band around the equator, and two orthogonal polar grids based on modified stereographic coordinates. Here, we generalise this by letting an equatorial band be covered with a reduced grid system, which can decrease the total number of grid points by at least 20%, with no substantial change in accuracy. Centred finite differences of high order are used for the spatial discretisation of the underlying differential equations and the explicit fourth order Runge-Kutta method for the integration in time. In the paper mentioned above, we demonstrate accuracy in the total mass, which is much higher than needed for NWP, for smooth solutions. Here we show that this holds also for non-smooth solutions, by considering the Cosine bell no. 1 and the mountain problem no. 5 in Williamson et\ua0al., the solutions of which have discontinuous second and first order derivatives, respectively