The rapidly increasing computing powers allow global atmospheric simulations with aggressively high resolutions, which challenges traditional model design principles. This paper presents a Low Mach number Approximate Riemann Solver (LMARS) based unstaggered finite-volume model for solving the shallow-water equations on arbitrary gnomonic cubed-sphere grids. Using a novel reference line-based grid-generation process, it unifies the representation of arbitrary gnomonic cubed-sphere grid projections and permits high-efficiency 1D reconstruction in the halo regions. The numerical discretization also extends a widely used pressure gradient algorithm with the LMARS viscous term, thus improves the model’s stability for various numerical applications. The solver demonstrates a broad range of organic diffusion control without any explicit filters, validated by a comprehensive set of test cases. Lastly, a newly introduced splash on the sphere test verifies the solver’s desirable dispersion properties and consistent performance among different grid types. This study paves a solid foundation for a new generation of global circulation models with kilometer horizontal scales.