Abstract
Abstract. In this paper, an efficient and conservative collocation method is proposed and used to develop a global shallow-water model. Being a nodal type high-order scheme, the present method solves the pointwise values of dependent variables as the unknowns within each control volume. The solution points are arranged as Gauss–Legendre points to achieve high-order accuracy. The time evolution equations to update the unknowns are derived under the flux reconstruction (FR) framework (Huynh, 2007). Constraint conditions used to build the spatial reconstruction for the flux function include the pointwise values of flux function at the solution points, which are computed directly from the dependent variables, as well as the numerical fluxes at the boundaries of the computational element, which are obtained as Riemann solutions between the adjacent elements. Given the reconstructed flux function, the time tendencies of the unknowns can be obtained directly from the governing equations of differential form. The resulting schemes have super convergence and rigorous numerical conservativeness. A three-point scheme of fifth-order accuracy is presented and analyzed in this paper. The proposed scheme is adopted to develop the global shallow-water model on the cubed-sphere grid, where the local high-order reconstruction is very beneficial for the data communications between adjacent patches. We have used the standard benchmark tests to verify the numerical model, which reveals its great potential as a candidate formulation for developing high-performance general circulation models.
Highlights
A recent trend in developing global models for atmospheric and oceanic general circulations is the increasing use of the high-order schemes that make use of local reconstructions and have the so-called spectral convergence
We introduce a new scheme which is different from the existing nodal discontinuous Galerkin (DG) and spectral element (SE) methods under the flux reconstruction (FR) framework
A three-point high-order Gauss–Legendrepoint-based conservative collocation (GLPCC) scheme is proposed under the framework of flux reconstruction
Summary
A recent trend in developing global models for atmospheric and oceanic general circulations is the increasing use of the high-order schemes that make use of local reconstructions and have the so-called spectral convergence. 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 (Chen and Xiao, 2008; Li et al, 2008; Ii and Xiao, 2010; Chen et al, 2014b). A global shallowwater equation (SWE) model has been developed by implementing the three-point GLPCC scheme on a cubed-sphere grid. The extension of the proposed scheme to a global shallow-water model on a cubed-sphere grid is discussed in Sect.
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