Application of the approximate Riemann solver to vertical motion of air in nonhydrostatic atmospheric models

An application study of three kinds of vertical approximate Riemann solvers have been carried out based on a multi-moment nonhydrostatic atmospheric model, which has the characteristics of high accuracy and numerical conservation. The conservative finite difference scheme is used in the vertical direction and the numerical flux in the cell boundary is realized by solving the Riemann problem, which plays a key role in accurately simulating vertical motion in the nonhydrostatic atmosphere. LLF (Local Lax-Friderich), LMARS (Low Mach Approximate Riemann Solver) and HLLC (Harten-Lax-van Leer Contact) are three kinds of approximate Riemann solvers commonly used in the computational fluid dynamics (CFD), and their computational cost and complexity are gradually increasing. One-dimensional standard numerical test show that the cost of LLF solver is the lowest, yet it has strong dissipation. LMARS is assumed to be suitable for atmospheric flow, and its numerical viscosity is not so large and the cost of computation is modest. The inclusion of the third wave in HLLC can avoid excessive numerical dissipation of the intermediate characteristic field. By adjusting the coefficient of the largest eigenvalue of different eigenwaves in LLF solver, the optimized LLF solver can achieve the same performance as that by the relatively complex LMARS and HLLC approximate Riemann solvers, and remain the lowest computational cost. Two-dimensional nonhydrostatic numerical test indicate that the optimized LLF approximate Riemann solver correctly simulates small-scale nonhydrostatic vertical motion and is competitive with the more complex LMARS and HLLC approximate Riemann solvers without increasing the amount of computation. This result provides a good reference for the study of nonhydrostatic atmospheric numerical models.