Numerical modeling of atmospheric phenomena, characterized as complex, nonlinear, and chaotic dynamical systems, requires accurate, efficient, and robust numerical methods. The shallow-water equations, as a simplified surrogate model of atmospheric dynamics, provide a fundamental framework for developing and benchmarking numerical schemes. However, their solution on the sphere is challenging due to the presence of polar singularities and nonuniform grid-point clustering in conventional latitude–longitude grids, which impose severe time-step restrictions and may cause numerical instability. In addition, conventional schemes may struggle to balance high-order accuracy and long-term stability.

This study develops and implements a numerical framework based on the shallow-water equations and the Yin–Yang overset grid, which eliminates polar singularities, alleviates the Courant–Friedrichs–Lewy (CFL) time-step restriction, and maintains approximately uniform horizontal resolution. The key innovation is the application of a fourth-order compact MacCormack (CMC4/2) scheme for spatial discretization on two Yin–Yang grid configurations under orographic forcing. The CMC4/2 scheme provides high-order spatial accuracy and favorable agreement with a spectral reference solution in resolving fine-scale atmospheric structures. Planetary-scale mountain-wave responses are also simulated and compared with those obtained using the classical second-order MacCormack (MC2) scheme. Time integration employs the classical fourth-order Runge–Kutta (RK4) method for the fourth-order spatial discretization and a second-order RK time-integration scheme for the lower-order scheme. Bilinear and bicubic interpolation methods are used for data exchange between the Yin and Yang subgrids.

The model is validated using Williamson et al. (1992) Test Case 5, “Zonal flow over an isolated mountain,” and Zerroukat and Allen (2015) Test Case 3, “Midlatitude jets over two isolated mountains”. The performance of the algorithms developed is evaluated by comparing the height field with the reference solution and conservation of mass, total energy, and potential enstrophy, together with consideration of numerical stability and computational cost. The results show that CMC4/2 controls dispersive errors much more effectively than MC2. During long-term integrations, MC2 exhibits noticeable losses of mass and energy and spurious oscillations in potential enstrophy, whereas CMC4/2 largely remains devoid of these deficiencies. On the modified Yin–Yang grid (MYY), CMC4/2 provides better conservation of mass and energy, a cleaner solution with suppressed spurious oscillations in potential enstrophy, and lower computational cost than on the modified Yin–Yang grid with identical component grids (MYYIC), making CMC4/2–MYY the most robust and stable configuration investigated. Its computational complexity, estimated by power-law regression, is characterized by a power law with the exponent p = 0.93 and R² = 0.97.

The interpolation scheme at overlapping boundaries proves to be critical for the accuracy and stability of the overset-grid method. Although bilinear interpolation is less expensive, bicubic interpolation is essential for preserving high-order accuracy, limiting the propagation of interpolation errors into the interior, and suppressing spurious wave generation, particularly during long-term integrations. Without numerical filtering, CMC4/2 combined with bicubic interpolation on the Yin–Yang grid successfully reproduces the sequence of nonlinear Rossby waves generated downstream of the mountain, with favorable agreement with the spectral reference solution. Also, the results demonstrate that the Yin–Yang grid alleviates the geometric and numerical limitations of conventional latitude–longitude grids while improving conservation, numerical stability, and computational efficiency.