Next generation weather prediction atmospheric models will have horizontal resolution of about 3-5 km. The problem dimension will be 1010. One will need to use efficiently 104-105 computational cores to make a practical operational forecast. This leads to the need for the deep revision of numerical methods and algorithms used in atmospheric models. One of the problems to be solved is the horizontal discretization of atmospheric dynamics equations on the quasi-uniform spherical grids. This problem can be investigated using shallow water model that is much computationally cheaper than the use of full atmosphere model. We are developing an atmospheric dynamics solver for the next generation numerical weather prediction model at the Institute of Numerical Mathematics and Hydrometeorological center of Russia. Within this work, the solver for the shallow water equations using gnomonic cubed-sphere grid has been developed. The solver is verified using standard shallow water test cases. The accuracy of the presented solver is analysed. The good agreement to the reference solutions is achieved, when 4-th order spatial approximations are used.