  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.
