A Study on the Precision of Runge-Kutta Method with Various Orders of Li Difference Scheme
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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。-
关键词:
- Runge-Kutta-Li格式 /
- 高阶算法 /
- Burgers方程
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.-
Key words:
- Runge-Kutta-Li scheme /
- High-order /
- Burgers' equation
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图 1 RKL（Runge-Kutta-Li）方法的线性平流试验：（a）Gauss波初始场；（b）计算误差随空间精度阶数的变化。（b）中横坐标为空间精度阶数，纵坐标为误差取对数（log10），粉、蓝、红、绿、黑色分别代表时间精度为2、3、4、5、6阶
Figure 1. The experiments of linear advection equation by RKL (Runge-Kutta-Li) method: (a) Gaussian-type initial condition; (b) error versus spatial difference order. In (b), the abscissa is the spatial difference order, the ordinate is the logarithm of the error (log10), and the purple, blue, red, green and black curves denote the second-order, third-order, fourth-order, fifth-order and sixth-order time-integration schemes, respectively
图 2 RKL方法的无粘性Burgers方程试验：（a）t=0时的初始场；（b）t=0.8时u的理论解；（c）计算误差随空间精度阶数的变化。（c）中横坐标为空间精度阶数，纵坐标为误差取对数（log10），粉、蓝、红、绿、黑色分别代表时间精度为2、3、4、5、6阶
Figure 2. The experiments of the nonlinear Burgers' equation by RKL method: (a) Initial u at t=0; (b) analytical solution of u at t=0.8; (c) error versus spatial difference order. In (c), the abscissa is the spatial difference order and the ordinate is the logarithm of error (log10); purple, blue, red, green and black curves correspond to the second-order, third-order, fourth-order, fifth-order and sixth-order time-integration schemes, respectively
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