The aim of this paper is to design the explicit radial basis function (RBF) Runge-Kutta methods for the initial value problem. We construct the two-, three- and four-stage RBF Runge-Kutta methods based on the Gaussian RBF Euler method with the shape parameter, where the analysis of the local truncation error shows that the s-stage RBF Runge-Kutta method could formally achieve order s+1. The proof for the convergence of those RBF Runge-Kutta methods follows. We then plot the stability region of each RBF Runge-Kutta method proposed and compare with the one of the correspondent Runge-Kutta method. Numerical experiments are provided to exhibit the improved behavior of the RBF Runge-Kutta methods over the standard ones.

翻译：本文旨在针对初值问题设计显式径向基函数（RBF）龙格-库塔方法。我们基于含形状参数的高斯RBF欧拉方法，构造了两级、三级和四级RBF龙格-库塔方法，其中局部截断误差分析表明，s级RBF龙格-库塔方法理论上可达到s+1阶精度。随后给出了这些RBF龙格-库塔方法的收敛性证明。接着，我们绘制了所提出的每种RBF龙格-库塔方法的稳定区域图，并将其与对应标准龙格-库塔方法的稳定区域进行了比较。数值实验展示了RBF龙格-库塔方法相对于标准方法的性能改进。