Shallow water model on cubed-sphere by multi-moment finite volume method

Abstract

A global numerical model for shallow water flows on the cubed-sphere grid is proposed in this paper. The model is constructed by using the constrained interpolation profile/multi-moment finite volume method (CIP/MM FVM). Two kinds of moments, i.e. the point value (PV) and the volume-integrated average (VIA) are defined and independently updated in the present model by different numerical formulations. The Lax-Friedrichs upwind splitting is used to update the PV moment in terms of a derivative Riemann problem, and a finite volume formulation derived by integrating the governing equations over each mesh element is used to predict the VIA moment. The cubed-sphere grid is applied to get around the polar singularity and to obtain uniform grid spacing for a spherical geometry. Highly localized reconstruction in CIP/MM FVM is well suited for the cubed-sphere grid, especially in dealing with the discontinuity in the coordinates between different patches. The mass conservation is completely achieved over the whole globe. The numerical model has been verified by Williamson's standard test set for shallow water equation model on sphere. The results reveal that the present model is competitive to most existing ones.

High-order central ENO finite-volume scheme for ideal MHD

“…In recent years, cubed-sphere grids have gained increasing popularity for simulating fluid flow in domains between concentric spheres, first in the area of climate and weather modelling [18,19,20,21,22,23,24], but more recently also in areas like astrophysics [25,26]. Very recently, Ivan et al [14,15] have proposed a second-order parallel solution-adaptive computational framework for solving hyperbolic conservation laws on 3D cubed-sphere grids and applied the formulation to the simulation of several magnetized and nonmagnetized space-physics problems.…”

A high-order conservative collocation scheme and its application to global shallow-water equations

“…As in the above case, to compare with our former fourth-order model this test case is checked on grid G 20 having the similar DOFs as the former 32×32×6 grid. The conservation errors are −9.288 × 10 −7 for total energy and −1.388 × 10 −5 for potential enstrophy and much smaller than those by fourth-order model in Chen and Xiao (2008).…”

“…Numerical results of height fields and absolute errors are shown in Fig. 8 for tests on grid G 12 , which means there are 12 elements in both ξ and η directions on every patch, in the different flow directions, i.e., γ = 0 and γ = and l ∞ = 8.045 × 10 −7 , which are almost 1 order of magnitude smaller than those on grid 32×32×6 (with similar number of DOFs; 256 DOFs along the equator) in Chen and Xiao (2008). The influence of patch boundaries on the numerical results can be found in the plots of the absolute errors.…”

“…Furthermore, a side effect of this choice is that the discontinuous coordinates are found along the boundary edges between adjacent patches. In Chen and Xiao (2008), we have shown that the compact stencils for the spatial reconstructions through using local DOFs are beneficial to suppress the extra numerical errors due to the discontinuous coordinates. …”

“…A FR scheme solves the values at the solution points located within each grid element, and the volume-integrated values, which are the weighted summation of the solutions, can be numerically conserved. We recently proposed a class of local high-order schemes, named multi-moment schemes, which were used to develop the accurate shallow-water models on different spherical grids Li et al, 2008;Ii and Xiao, 2010;Chen et al, 2014b). By introducing a multi-moment concept, we showed in Xiao et al (2013) that the flux reconstruction can be implemented in a more flexible way, and other new schemes can be generated by properly choosing different types of constraint conditions.…”