CENTRAL DIFFERENCE SCHEMES WITH HIGH ORDER ACCURACY ON NONUNIFORM GRIDS: THEORETICAL DERIVATION AND IDEAL TEST
-
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。Abstract: 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 2nd, 4th, and 6th 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 2nd order to the 6th order. It is also found that higher accuracy schemes have advantages in regions of steep gradient or coarser resolution, in which the errors of 4th and 6th order schemes are much smaller than that of the 2nd order scheme.
-
表 1 2阶、4阶和6阶非均匀网格差分方案的数值解的误差L2
误差L2 2阶 4阶 6阶 1.5 s 6.57×10-4 2.65×10-5 2.54×10-5 2.0 s 4.15×10-4 5.91×10-5 5.79×10-5 2.5 s 3.31×10-4 9.90×10-5 1.00×10-4 
下载: 导出CSV
表 2 2阶、4阶和6阶非均匀网格差分方案的平流方程数值解的误差L2
误差L2 2阶 4阶 6阶 1 s 2.52×10-3 2.97×10-4 2.96×10-4 3 s 8.86×10-3 3.26×10-3 1.55×10-3 5 s 3.55×10-2 1.58×10-2 9.51×10-3 10 s 8.43×10-2 4.67×10-2 3.39×10-2
下载: 导出CSV
-
[1] CULLEN M J P, DAVIES T. A conservative split-explicit integration scheme with fourth-order horizontal advection[J]. Q J R Meteor Soc, 1991, 117(501): 993-1 002 [2] CROWELL S, WILLIAMS D, MAVRIPLIS C, et al. Comparison of traditional and novel discretization methods for advection models in numerical weather prediction[C]. Computational Science-iccs, International Conference, Baton Rouge, LA, USA, May, Part Ⅱ. Springer-Verlag, 2009: 263-272. [3] LESLIE L M, PURSER R J. High-order numerics in an unstaggered three-dimensional time-split semi-lagrangian forecast model[J]. Mon Wea Rev, 1991, 119(7): 1 612-1 623. [4] TREMBACK C J, POWELL J, COTTON W R, et al. The forward-in-time upstream advection scheme: extension to higher orders[J]. Mon Wea Rev, 1987, 115(2): 540-555. [5] KENT J, WHITEHEAD J P, JABLONOWSKI C, et al. Determining the effective resolution of advection schemes. Part Ⅰ: Dispersion analysis[J]. J Comp Phys, 2014, 278(C): 497-508. [6] 王明欢, 沈学顺, 肖锋. GRAPES模式中高精度正定保形物质平流方案的研究Ⅱ:连续实际预报试验[J].气象学报, 2011, 69(1): 16-25. [7] MORINISHI Y, LUND T S, VASILYEV O V, et al. Fully conservative higher order finite difference schemes for incompressible flow[J]. J Comp Phys, 1998, 143(1): 90-124. [8] HAM F E, LIEN F S, STRONG A B. A fully conservative second-order finite difference scheme for incompressible flow on nonuniform grids[J]. J Comp Phys, 2002, 177(1): 117-133. [9] 田芳, 田振夫.非均匀网格上求解对流扩散问题的高阶紧致差分方法[J].宁夏大学学报(自然科学版), 2009, 30(3): 11-14 [10] CHU P C, FAN C. Sixth-order difference scheme for sigma coordinate ocean models[J]. J Phys Ocean, 1997, 27(9): 2 064-2 071 [11] CHU P C, FAN C. An accuracy progressive sixth-order finite-difference scheme[J]. J Atmos Ocean Techn, 2001, 18(7): 1 245-1 257. [12] MCCALPIN J D. A comparison of second-order and fourth-order pressure gradient algorithms in a σ - co-ordinate ocean model[J]. International Journal for Numerical Methods in Fluids, 1994, 18(4): 361-383. [13] KÁLNAY E. On the use of nonuniform grids in finite-difference equations[J]. J Comp Phys, 1972, 10(2): 202-210. [14] 姜勇强, 张维桓, 周祖刚, 等.不等距分层中尺度η坐标暴雨模式的设计及数值试验[J].热带气象学报, 2001, 17(3): 301-307 [15] 孙建安, 贾伟, 吴广智.一种非均匀网格上的高精度紧致差分格式[J].西北师范大学学报(自然科学版), 2014, 50(4): 31-35, 40. [16] 徐道生, 陈德辉.非均匀分层下的二阶精度垂直差分方案及其在GRAPES模式中的应用[J].大气科学, 2020, doi: 10.3878/j.issn.1006-9895.1906.19145. [17] ALPER K, IDRIS D. Cubic B‐spline differential quadrature methods and stability for burgers' equation[J]. Engineering Computations, 2013, 30(3): 320-344. [18] DOGAN A. Numerical solution of RLW equation using linear finite elements within Galerkin's method[J]. Applied Mathematical Modelling, 2002, 26(7): 771-783. [19] 李五明.对流方程差分格式稳定性判定[J].河南理工大学学报(自然科学版), 2012, 31(3): 369-372. [20] 薛纪善, 陈德辉.数值预报系统GRAPES的科学设计与应用[M].北京:科学出版社, 2008: 132-136. [21] 刘宇, 曹建文.适用于GRAPES数值天气预报软件的ILU预条件子[J].计算机工程与设计, 2008, 29(3): 217-220. [22] 朱文达, 陈子通, 张艳霞, 等.高分辨地形对华南区域GRAPES模式地面要素预报影响的研究[J].热带气象学报, 2019, 35(6): 801-811. [23] 张诚忠, 薛纪善, 黄燕燕, 等.资料同化对2017年登陆广东沿海台风的短期降水与路径预报影响[J].热带气象学报, 2019, 35(5):577- 586. [24] 陈子通, 戴光丰, 钟水新, 等.中国南海台风模式(TRAMS-v2.0)技术特点及其预报性能[J].热带气象学报, 2016, 32(6):831-840. -
点击查看大图
图(7) / 表(2)
计量
- 文章访问数: 236
- HTML全文浏览量: 18
- PDF下载量: 18
- 被引次数: 0
粤公网安备 4401069904700003号