Single-Step Sixth Stage Implicit Runge-Kutta Method for First-Order Initial Value Problems of Ordinary Differential Equations

Abstract

We present a single-step block hybrid method obtained through multistep collocation (MC) approach, by incorporating three off-grid interpolation and two off-grid collocation points in a continuous linear hybrid method as opposed to a similar work where two-step methods were obtained. The discrete hybrid methods are used in the formulation a block method which is then reformulated into a new single-step sixth-stage implicit Runge-Kutta method (SIRK) for the numerical treatment of first-order ordinary differential equations. The basic convergence and A-stability properties of the new method are established. Numerical experiments performed using the new method further revealed a reduction in the absolute errors, showing that this method is a better candidate for similar existing ones in the literature, as such it should be used for such class of problems.