To achieve strong scalability in global models and accurately simulate atmospheric flows over complex underlying surfaces, unstructured grids have attracted increasing attention due to their flexibility. However, developing high-order numerical models on these complex grids remains a major challenge. In this study, we propose a multi-moment scheme for three-dimensional nonhydrostatic atmospheric dynamics on a hexagonal prism grid. Two kinds of moments, namely the point value and the volume-integrated average, are employed to implement the spatial reconstruction using a single-cell stencil, enabling robust and accurate formulation on unstructured grids. The quadratic reconstruction polynomial is determined by a least-squares approach to minimize asymmetric errors. In the vertical direction, a conservative finite-difference method is employed within a height-based terrain-following coordinate system. Benchmark tests demonstrate that the model achieves third-order accuracy and accurately simulates complex three-dimensional atmospheric flows. The proposed scheme can be readily extended to the icosahedral–hexagonal grid to develop a global atmospheric model. Furthermore, this study establishes a unified multi-moment framework for constructing high-order atmospheric models on fully unstructured meshes.

In this study, the adaption of a novel three-point multi-moment constrained finite-volume transport scheme for uniform points with center constraints (MCV3_UPCC) to cubed sphere geometry is implemented and described. For the MCV3_UPCC scheme, the three equidistant solution points are located within a single cell and a polynomial of 4th degree can be built by imposing the multi-moment center constraints. The resultant scheme has third-order accuracy and guarantees the exact numerical conservation. The Fourier analysis of MCV3_UPCC scheme demonstrates that the novel MCV3_UPCC has better numerical dissipation and dispersion than the original 3rd order Multi-moment Constrained finite Volume (MCV3) scheme. Then it is applied to quasi-uniform cubed-sphere grid, which is designed to avoid the polar problem on the traditional latitude–longitude grid. To suppress the non-physical numerical oscillations, a bound-preserving (BP) algorithm to constrain the conserved advected tracer to within the initial maxi-mum and minimum values is also implemented. The scheme is validated with several widely used benchmarks involving prescribed non-divergent two-dimensional flow on the sphere and different initial tracer distributions. The resulting conservative transport model with high-order accuracy and positive preserving property is comparable to other high-order schemes and has the potential for the numerical simulation of various traces in the atmosphere.