Summation-by-parts finite-difference shallow water model on the cubed-sphere grid. Part I: Non-staggered grid

Abstract

Summation-by-Parts Finite Differences (SBP-FD) is an approach for approximation of differential operators satisfying a discrete analogue of integration by parts analytic property. SBP-FD allows one to build provably stable high-order spatial approximations. SBP-FD methods are widely used for approximation of partial differential equations in multiblock domains with logically-rectangular curvilinear mesh inside. The gnomonic cubed-sphere grid is an example of such a domain. However, applications of the SBP-FD approximation for the cubed-sphere grid in meteorological context are not a widespread research area.We present a SBP-FD based shallow water model using a non-staggered grid. The model is total-energy conserving, mass-conservative and has discrete analogues of other mimetic properties such as curl-free gradient property. The shallow water model is tested with the commonly used Williamson test suite supplemented with the Galewsky barotropic instability case. High-order convergence is shown for tests with analytic solutions. The SBP-FD shallow water model is more accurate than low-order mimetic finite-element counterparts, but slightly less accurate than high-order finite-volume, spectral-elements and discontinuous Galerkin schemes.