High Order Runge-Kutta-Li Scheme to Solve Two-dimensional Linear Advection Equations

In this study, aiming to take full advantages of Li’s high-order spatial differential method (Li, 2005), we implement the hybrid Runge-Kutta-Li (RKL) scheme to solve a two-dimensional (2D) linear advection equation. The results indicate that the computation error increased linearly with time. The experiments with a no-rotate background field of the Gaussian initial values by RKL scheme could obtain a precise result. The effective spatial orders were 5, 7, 9, and 10, corresponding to temporal orders of 3, 4, 5, and 6, respectively. The fifth- (sixth-) order Runge-Kutta (RK) integration scheme with the ninth- (tenth-) order Li’s difference scheme in spatial direction controlled the error within 10^-7(10^-8). The effective order of Li’s scheme (Li, 2005) tended to increase with the increase in the RK order, and the total error gradually decreased.Another rotated background field integrated from an eccentric Gaussian-type initial featured similar results. The effective spatial order was 10 when a third-order RK scheme was applied, and they increased to 16, 22, and 22 when the order of RK scheme changed to 4, 5, and 6, respectively. The computation error could be controlled within 10^-15-10^-16, and the peak of Gaussian initial values were well maintained. The error decreased sharply while the order of RKL scheme increased, which indicates that the RKL is very effective to deal with such problem. The experiments of RKL scheme to solve a cone initial case (with rotated background filed) indicate that the fourth, fifth, and sixth RK integration obtained almost the same precision result as the third-order RK scheme. The high-order scheme was not as effective as it was for the Gaussian initial condition when it addressed a problem that had discontinuous derivates. The computed solution was not positive in the whole grids, and in some places, the error was downward, while it was upward in some other places. The increase of spatial order could make the error smaller, but the error descent was not very sharp. This result suggests that the high-order spatial difference scheme has some benefits, but we should not expect that an ultra-high-order scheme will lead to an ultra-high precise result. This phenomenon reveals that the high-order scheme is limited by the continuous property of the initial condition, and as a result, the error order is directly proportional to the order of derivates of the initial condition. The proper boundary conditions are important for the above computation cases when the RKL scheme is applied. For instance, in the computation of the rotated background cases, the value outside the grid box tends to 0 at , which is a feasible boundary condition. An improper boundary condition may cause the computation to be unstable, or the error cannot be controlled to an acceptable range.