Comparison of numerical solution of the spherical two-dimensional advection equation using three types of Yin–Yang grid

Due to the approximately spherical nature of the earth and the complex nature of atmospheric and oceanic flows, numerical solution of corresponding governing equations requires using an appropriate coordinate system on the spherical geometry. All spherical grids defined for the spherical surface or shell, have their own advantages and disadvantages generally. The 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 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 for the conventional latitude–longitude grid; 6- We can adapt the existing latitude–longitude discretizations and codes for the use with the Yin–Yang grids. In this research, three types of the Yin–Yang grid are compared: the rectangular (basic), modified and modified with identical components. It is worth noting that the Yin–Ying grid with identical components is introduced for the first time in the present study. The central second-order finite difference scheme is applied to solve the two-dimensional advection equation on three types of Yin–Yang grid for a well-known test case. In addition, the fourth-order Runge–Kutta method is used to advance the governing equation in time. 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 and modified Yin–Yang grids entails a lower computational cost than the modified Yin–Yang grid with identical components. In addition, global errors are computed using the absolute, square and infinite norms. By calculating the errors using these norms, it can be seen that there is a slight increase in the errors in all three grids compared to the conventional latitude–longitude grid. Another point to note is a little higher accuracy of modified Yin–Yang grid with identical components relative to rectangular and modified Yin–Yang grids in the same resolution; though, the higher accuracy is associated with a relative increase in computational cost. In the considered algorithm, reduction of the accuracy in using Yin-Yang grids is likely due to interpolation. However, interpolation is an essential part of numerical solving process for various oceanic and atmospheric equations on Yin-Yang grids in spherical geometry.