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

Study on the precision of Runge-Kutta method with various order Li difference scheme
摘要点击 242  全文点击 44  投稿时间:2017-09-16  修订日期:2018-03-29
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基金:  
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wangpengfei中国科学院大气物理研究所
楚苹瓖中国科学院大气物理研究所
周任君中国科学技术大学
黄刚Huang Gang中国科学院大气物理研究所
引用:王鹏飞,楚苹瓖,周任君,黄刚.2018.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
Citation:wangpengfei,Huang Gang.2018.Study on the precision of Runge-Kutta method with various order Li difference scheme[J].Chinese Journal of Atmospheric Sciences (in Chinese)
中文摘要:
      为了充分发挥高阶Li空间微分方案的优点,实现了时间积分为2-6阶Runge-Kutta(RK)格式的偏微分方程求解算法(RKL)。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、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(RK4-6)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 taking full advantages of Li’s high order spatial differential method. A set of numerical experiments are done to analysis how the computation error are affected by the order of integration scheme. The results of linear advection equation indicates that with the square-wave type initial values the scheme can only obtain a third-order accuracy result, but for the Gaussian function type of initial values, the scheme can obtain a better result. The 5(6) order Runge-Kutta (RK) integration scheme correspond to 9(10) order Li’s difference scheme in spatial direction and the total error can be controled within10-7(10-8). The order of Li’s scheme tends to increase while the RK order increases, meanwhile the total error gradually decrease. When we compute the nonlinear Burgers equation, whether RKL scheme can obtain good results are not only depend on the form of initial filed, but is related to the target computation time. When the derivative is continuous (and does not appear infinite value) at the target observation time, 4-6 order RKL scheme is effective. On the contrary, if the derivative is discontinuous, or the derivative is tend to infinity, and then the RKL scheme can’t obtain very high precision numerical solution. In the case (Burgers with smooth initial), the order of Li’s scheme is increase while the RK order increased, but the relation between them shows a nonlinear tendency (which can be specific through some fitting methods). The results indicate that when the order of time integral is more than 3, the corresponding optimal spatial difference order can be higher than 6, which again validates the finding of previous study that when the order of spatial difference above 6 gets no improvement is attributed to none high enough time integral scheme. Compare with Taylor-Li(Wang, 2017) scheme, the 5-6 order RKL scheme is more easy in program coding and more precise than the 3 order scheme. To conclude, the high order RKL scheme can be applied to some complicated type of partial differential equations and is valuable to many other similar computation cases.
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