Positive Definite Advection Transport Algorithm for Conservation Law Equations on Nonuniform Irregular Grids

The multidimensional positive definite advection transport algorithm (MPDATA) is an important numerical method for the computation of atmospheric dynamics. MPDATA is second-order accurate, positive definite, conservative, and computationally efficient. However, the method is problematic in which it results in a loss of precision when computing a nonuniform irregular grid. Furthermore, research revealed two reasons for this problem. On the one hand, numerical discretization of boundary derivatives of the finite-volume method is incompatible with nonuniform meshes (or grids); on the other hand, the up-wind scheme of staggered grids is not applicable to the calculation of irregular grids. We overcome these two problems by using the multipoint Taylor expansion method to obtain a boundary derivative numerical approximation scheme that does not depend on the grid structure. Furthermore, combined with the well-balance central-upwind scheme, a positive definite advection scheme for irregular meshes is proposed. Then, the positivity of the new numerical scheme is analyzed. Finally, the result of this study is verified by numerical simulation.