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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. [ABSTRACT FROM AUTHOR]