TWO-STEP RUNGE-KUTTA METHODS FOR THE NUMERICAL INTEGRATION OF NONLINEAR SINGULAR AND SINGULARLY PERTURBED IVPs

TWO-STEP RUNGE-KUTTA METHODS

Abstract

We construct stable Runge-Kutta methods with fewer function evaluations per step by using the two end points of the integration interval as collocation points in addition to the Gaussian interior collocation points. As a result, the methods' rate of convergence is quite high. The advantage of these methods as compared, for example, with methods of the conventional type (Gauss, Radau and Lobatto Runge-Kutta methods) consists of the fact that they provide uniform approximations of the solution of singularly perturbed systems in ordinary differential equations (ODEs), over the entire integration interval. This is in contrast to the conventional Runge-Kutta methods for which the continuous approximation to the exact solution of ODEs is obtained at the mesh points only. Although the computational cost of these methods is little more than the explicit methods, the advantages gained such as high orders, improved regions of absolute stability and lower error constants, make the methods suitable for solving stable systems. By demonstrating two potential implementations of the collocation methods, the desired goal of the derivation is achieved. Results obtained are presented in Tables, while graphic surface curves are shown in Figures to illustrate the accuracy and effectiveness of the derived methods.