Numerical solution of two-dimensional advection equation in spherical geometry using the fourth-order compact MacCormack scheme on a Yin-Yang grid

Due to the approximately spherical nature of the atmosphere, oceans and other layers of the Earth and the complex nature of atmospheric and oceanic flows, numerical solution of their governing equations requires using an appropriate coordinate on the spherical geometry. All spherical grids defined for the spherical surface or shell, generally have their own advantages and disadvantages. In general, it can be said that there is no spherical grid which has all the following characteristics: 1- The grid is orthogonal; 2- There is no singularity; 3- There is no grid convergence problem; and defined over entire spherical surface. Thus, we have to discard one of these incompatible conditions. An overset grid is a type of grid that divides a spherical surface into subregions. Yin–Yang grid belongs to the family of overset grids. This coordinate is composed of two grid components named Yin and Yang with partial overlapping at their boundaries. Some of the advantages of the Yin–Yang grids are as follows: 1- Yin and Yang grid components are both orthogonal and based on the conventional latitude–longitude grid; 2- The singular points are absent; 3- The metric factors of the both grid components are analytically known; 4- The grid lengths are uniform approximately; 5- It requires less grid points than the conventional latitude–longitude grid; and 6- We can adapt the available latitude–longitude discretization and codes for the use with the Yin–Yang grids. In addition, we have to use interpolation for setting boundary conditions for the two grid components. The interpolation scheme that has been used in this study is bilinear. In this research, a type of the Yin–Yang grid called the rectangular (basic) is applied to solve the two-dimensional advection equation for a well-known test case using the fourth-order compact MacCormack scheme. Furthere, the fourth-order Runge–Kutta method is used for time stepping. Results show that using the Yin–Ying grids to solve the advection equation is highly effective in reducing the computational cost compared to the conventional latitude–longitude grid, however the use of rectangular Yin–Yang grid entails a lower accuracy than the conventional latitude–longitude grid. In this numerical test, global errors are computed using the absolute-value, Euclidean and maximum norms. By calculating the errors using these norms, there is an order of magnitude increase in the errors in rectangular Yin-Yang grid compared to the conventional latitude–longitude grid. This increase in error can come from the inevitable interpolation process involved in the Yin-Yang grid.