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A numerical constraint was developed to improve the global and local conservation for the Yin‐Yang grid system, which has been known as one of the quasi‐uniform grids on a sphere. Two‐dimensional cubic mass distribution within an individual mesh was assumed, to describe the subgrid‐scale structure of local properties and to ensure high‐order‐accuracy mass flux specification for the Yin‐Yang boundary. A three‐point Multi‐moment Constrained finite Volume scheme, in cooperation with a fourth‐order Runge–Kutta scheme, was selected for numerical transport with the help of Boundary Gradient Switching for oscillation suppression. The new scheme was tested with a couple of idealized numerical experiments in advection and shallow‐water models on the Yin‐Yang grid to verify its performance. Numerical results confirmed the exact mass conservation in spherical advection problems, and the numerical convergence rate reached fourth order in both advection and shallow‐water models. Computational stability, shape‐preserving and numerical oscillation‐free properties were also revealed in the nonlinear testing problems.
A three‐dimensional positivity‐preserving and conservative transport model on a cubed‐sphere grid is presented using a multimoment finite‐volume (MMFV) method. The three‐point fourth‐order MMFV method with boundary gradient switching projection is utilized to ensure nonoscillatory numerical solutions in the horizontal, while the simple and computationally efficient semi‐Lagrangian piecewise rational method is used to allow a large time step in the vertical. By means of horizontal and vertical separation, a Strang‐type splitting technique is adopted to advance the numerical solutions in time. Non‐negative corrections are made to achieve positive‐definite preservation exactly. The resulting transport model is inherently conservative, high‐accuracy, and positive‐definite, and allows a large Courant number no matter how dense the vertical grid is. Various benchmark test cases, including three representative advection tests and an idealized terminator chemistry test, are conducted to validate the performance of the model presented. The numerical results indicate that the model is quite competitive in comparison with the existing models in the literature and looks very promising in real atmospheric simulations.
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