适用于非静力大气模式的近似黎曼求解器应用研究

An applied study of approximate Riemann solvers for nonhydrostatic atmospheric model

中文关键词:  近似黎曼求解器，守恒型有限差分方法，多矩约束有限体积方法，非静力大气模式

英文关键词:Approximate Riemann solver, Conservative finite difference scheme, Multi-moment constrained finite volume method, Nonhydrostatic atmosphere model

中文摘要:

      基于多矩非静力大气模式，开展了三类垂向近似黎曼求解器应用研究。[目的]多矩非静力大气模式具有高精度与数值守恒特性，其垂向采用守恒的有限差分格式进行数值离散，而网格单元边界通量计算是通过求解黎曼问题来实现的，因此采用合适的近似黎曼求解器对准确模拟非静力大气垂直运动显得十分关键。[资料和方法]LLF（Local Lax-Friedrich）、LMARS（Low Mach Approximate Riemann Solver）和HLLC（Harten-Lax-van Leer Contact）为计算流体力学中常用的三种近似黎曼求解器，它们计算代价和复杂性逐渐增加。[结果]一维标准数值试验表明：LLF计算最为经济，但具有较强的耗散；LMARS具有适用于大气流动的假设，对于数值粘性的控制较好且计算量不大；HLLC建立的三波模型可以避免对中间特征场的过度数值耗散。[结论]基于LLF近似黎曼求解器计算经济的特点，通过优化LLF近似黎曼求解器各特征波动的粘性系数，能够实现与LMARS、HLLC近似黎曼求解器相同的性能，且计算代价最优。二维非静力数值试验表明，优化的LLF近似黎曼求解器能够规避常规LLF近似黎曼求解器的数值耗散过大问题，正确模拟小尺度非静力垂直运动，达到更复杂的LMARS、HLLC近似黎曼求解器模拟效果且并未增加计算量，这为非静力大气数值模式提供了良好参考价值。

英文摘要:

      Based on the multi-moment nonhydrostatic atmospheric model, an applied study of three kinds of vertical approximate Riemann solvers have been carried out. The multi-moment nonhydrostatic atmospheric model has the characteristics of high accuracy and numerical conservation. The conservative finite difference scheme is used in the vertical, and the numerical flux in the cell boundary is realized by solving the Riemann problem which plays a key role in accurately simulating the vertical motion of 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, and their computational cost and complexity are gradually increasing. One-dimensional standard numerical experiments show that the cost of LLF solver is the lowest, but it has strong dissipation. LMARS has the assumption that it is 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 largest eigenvalue of different eigenwaves in LLF solver, the optimized LLF solver can achieve the same performance in comparison with the relatively complex LMARS and HLLC approximate Riemann solvers, and still remain the lowest computational cost. The two-dimensional nonhydrostatic numerical experiments indicate that the optimized LLF approximate Riemann solver correctly simulates the small-scale nonhydrostatic vertical motion and is competitive to the more complex LMARS and HLLC approximate Riemann solvers without increasing the amount of computation, which provides a good reference value for the nonhydrostatic atmospheric numerical model.