非均匀网格下的高阶精度中央差分格式:理论推导和理想试验  

CENTRAL DIFFERENCE SCHEMES WITH HIGH ORDER ACCURACY ON NONUNIFORM GRIDS: THEORETICAL DERIVATION AND IDEAL TEST

作　　者：吴凯昕 徐道生[1] 陈德辉 李梦婕[1] WU Kai-xin;XU Dao-sheng;CHEN De-hui;LI Meng-jie(Guangzhou Institute of Tropical and Marine Meteorology/Guangdong Provincial Key Laboratory of Regional Numerical Weather Prediction,CMA,Guangzhou 510641,China;Numerical Weather Prediction Center of CMA,Beijing 100081,China)

机构地区：[1]中国气象局广州热带海洋气象研究所/广东省区域数值预报重点实验室,广东广州510641 [2]中国气象局数值预报中心,北京100081

出　　处：《热带气象学报》2020年第3期389-400,共12页Journal of Tropical Meteorology

基　　金：广东省气象局科学技术研究项目(GRMC2019M07);国家重点研发专项课题(2018YFC1506900);广东省气象局科学技术研究项目(GRMC2018Q02)共同资助。

摘　　要：传统的高阶精度有限差分格式通常是在均匀网格的基础上推导得到的,在非均匀网格的情况下它会出现精度退化的问题。基于泰勒展开方法构造了一种适用于非均匀网格的2阶、4阶和6阶精度中央有限差分方案,利用Burgers方程和一维平流方程对新方案的性能进行测试,着重分析新方案对其误差大小及分布形态的改进效果。数值模拟结果表明:在非均匀网格下,提高差分方案的精度可明显减小数值解误差(降低了70%~88%),特别是当差分精度从2阶提高到4阶的时候。同时,高阶精度方案在梯度变化较大或者网格距较粗区域的模拟结果更有优势,4阶和6阶精度方案在以上区域的误差远小于2阶精度方案。方案可用于提高数值天气预报模式中非均匀分层模式的垂直差分计算精度。Traditional central finite difference schemes are derived on uniform grids. However, the precision of such schemes will decrease on non-uniform grids. This article derives the 2 nd, 4 th, and 6 th order central finite difference schemes using Taylor expansion method. Burgers equation and 1-D advection equation are numerically solved by the new schemes on non-uniform grids. Comparing the numerical solution errors, we find that higher accuracy schemes can reduce up to 70%~88% numerical errors,especially when the accuracy is improved from the 2 nd order to the 6 th order. It is also found that higher accuracy schemes have advantages in regions of steep gradient or coarser resolution, in which the errors of4 th and 6 th order schemes are much smaller than that of the 2 nd order scheme.

关 键 词：非均匀网格 中央差分格式 高阶精度 BURGERS方程 一维平流方程 

分 类 号：P456.7[天文地球—大气科学及气象学]