A Study on the Precision of Runge-Kutta Method with Various Orders of Li Difference Scheme

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.