An Implicit Runge-Kutta Method for Perturbed Ordinary Differential Equations

A novel approach is developed for computing the numerical solution of perturbed ordinary differential equations using implicit Runge-Kutta methods. At each step of the integration of the differential equations, the method uses an enhanced initial approximation for each stage and then solves the implicit equations using a modified Picard iteration. The new approach offers greater efficiency for computing numerical solutions to perturbed initial value problems.