doi:  10.3878/j.issn.1006-9895.1805.17238
Runge-Kutta算法与Li差分法不同阶数配合对计算精度影响研究

A Study on the Precision of Runge-Kutta Method with Various Orders of Li Difference Scheme
摘要点击 590  全文点击 332  投稿时间:2017-09-16  
查看HTML全文  查看全文  查看/发表评论  下载PDF阅读器
基金:  国家自然科学基金项目41530426、41375112,中国科学院“关键技术人才”项目
中文关键词:  Runge-Kutta-Li格式  高阶算法  Burgers方程
英文关键词:  Runge-Kutta-Li scheme  High-order  Burgers' equation
              
作者中文名作者英文名单位
王鹏飞WANG Pengfei中国科学院大气物理研究所季风系统研究中心, 北京 100190;中国科学院大气物理研究所大气科学和地球流体力学数值模拟国家重点实验室, 北京 100029
楚苹瓖CHU Pingxiang中国科学院大气物理研究所季风系统研究中心, 北京 100190;中国科学院大学, 北京 100049
王立志WANG Lizhi中国科学院东亚区域气候-环境重点实验室, 北京 100029
周任君ZHOU Renjun中国科学技术大学地球和空间科学学院, 合肥 230026
黄刚HUANG Gang中国科学院大气物理研究所大气科学和地球流体力学数值模拟国家重点实验室, 北京 100029
引用:王鹏飞,楚苹瓖,王立志,周任君,黄刚.2019.Runge-Kutta算法与Li差分法不同阶数配合对计算精度影响研究[J].大气科学,43(1):99-106,doi:10.3878/j.issn.1006-9895.1805.17238.
Citation:WANG Pengfei,CHU Pingxiang,WANG Lizhi,ZHOU Renjun,HUANG Gang.2019.A Study on the Precision of Runge-Kutta Method with Various Orders of Li Difference Scheme[J].Chinese Journal of Atmospheric Sciences (in Chinese),43(1):99-106,doi:10.3878/j.issn.1006-9895.1805.17238.
中文摘要:
      为了充分发挥高阶Li空间微分方案(Li,2005)的优点,实现了时间积分为2~6阶Runge-Kutta(简称RK)格式的偏微分方程求解算法(简称RKL算法)。然后通过多组数值试验,研究了时间积分阶数对计算误差的影响。线性平流方程的试验结果表明对于方波函数型初值,2、4、5和6阶RK算法能获得和3阶精度差不多的结果,而对于高斯函数型的初值,高阶RKL算法可以取得较好的计算效果。RK为5(6)阶时,对应的Li微分阶数可达9(10)阶,总误差控制在10-7(10-8)以内。随RK阶数增加Li微分有效阶数有增加的趋势,而总误差在逐渐减小。计算非线性无粘Burgers方程时,RKL算法能否获得好的计算结果,除了受初始场形式的影响,还与计算的目标时刻有关。当目标时刻解的各阶导数连续(且未出现无穷大数值时),高阶(RK为4~6阶)算法是有效的;若出现了导数间断、或导数为无穷大,就会碰到冲击波解类型的问题,此时高阶RK算法也无法获得很高精度的数值解。此非线性的算例中,Li微分阶数仍然随RK阶数增加而增加,但增加的趋势不是线性的,具体变化关系可以通过实验结果拟合而获得。研究发现时间积分方案阶数大于3之后,对应的最优空间差分精度阶数可以比6阶提高很多,这再次证明了以前研究中6阶以上空间差分格式对结果无改进的现象,是由于没有使用足够高精度的时间积分方案引起的。相比于Taylor-Li(Wang,2017)算法,5~6阶的RK方法编程和实现简单,计算结果的精度比3阶算法要提高很多,因此,它是一种能够对复杂方程适用的简易高阶算法方案,具有一定的实用价值。
Abstract:
      We implement the hybrid Runge-Kutta-Li (RKL) scheme for the purpose to take full advantage of Li's high order spatial differential method (Li, 2005). A set of numerical experiments has been conducted to analyze how the computation error is affected by the order of integration scheme. The results of the linear advection equation indicate that with the square-wave type initial values, the scheme can only obtain a third-order accuracy. However, for the Gaussian function type of initial values, the scheme can obtain a better result. The fifth (sixth) order Runge-Kutta (RK) integration scheme corresponds to 9th (10th) order Li's difference scheme in spatial direction and the total error can be controlled within 10-7 (10-8). The order of Li's scheme tends to increase while the RK order increases, and the total error gradually decreases. When we compute the nonlinear Burgers' equation, whether the RKL scheme can obtain good results is not only dependent on the form of the initial field, but also related to the target computation time. When the derivative is continuous (and infinite value does not appear) at the target observation time, 4th-6th order RKL scheme is effective. On the contrary, if the derivative is discontinuous, or the derivative tends to infinity, the RKL scheme cannot obtain high-precision numerical solution. In this case (Burgers' with smooth initial), the order of Li's scheme still increases while the RK order increases, but the relation between them shows a nonlinear tendency (which can be specified through some fitting methods). The results indicate that when the order of time integral is more than three, the corresponding optimal spatial difference order can be higher than six. This result confirms the finding of previous studies that the order of spatial difference above six makes no improvement to the results due to the lack of high-precision time integral scheme. Compared with Taylor-Li (Wang, 2017) scheme, the 5th-6th order RKL scheme is easier to program and can yield more precise results than the third-order scheme. To conclude, the high order RKL scheme can be applied to some complicated types of partial differential equations and is valuable for many other similar computation cases.
主办单位:中国科学院大气物理研究所 单位地址:北京市9804信箱
联系电话: 010-82995051,010-82995052传真:010-82995052 邮编:100029 Email:dqkx@mail.iap.ac.cn
本系统由北京勤云科技发展有限公司设计
京ICP备09060247号