Lobatto Implicit Sixth Order Runge-Kutta Method for Solving Ordinary Differential Equations with Stepsize Control

Abstract

A method for solving ordinary differential equations has been developed using implicit Runge-Kutta methods. The implicit Runge-Kutta methods used are based in two quadratures of Lobatto type. The first quadrature produces the prindpltl Runge-kutta method which is of sixth order, while the second quadrature produces a Runge-Kutta method of third order which is embedded in the former. Both implicit Runge-Kutta methods constitute the Lobatto embedding form of third and sixth orders with
four stages. The most important advantage of this method is that it is implicit only in the second and third stages, which reduces considerably the cost computer calculations. The Butcher notation is used here for the analysis of the studied methods. In order to solve, for each step, the system of non-linear equations in the implicit auxiliary variables k2 and k3, an explicit Runge-Kutta method of four stages and fourth order is defined for the same intermediate points, such as the implicit sixth order Runge-Kutta method. This explicit method estimates the inicial values for the aforementioned auxiliary variables, and then, an iterative method of the type "fixed point" is used to solve the system of non-linear equations for each step. With the third order Runge-Kutta method, an estimation of the local truncation error may be calculated using a comparison with the sixth order method. This aspect is Used to control the step size when tolerances for the relative and absolute global errors are specified. An algorithm is presented to do this step control automatically. The implicit method, as is exposed here, is really useful and has demonstrated to be efficient to solve hllge alld stilf systems of ordinary differential equations. Finally, convergence criteria and stability analysis are studied for the R.unge-Kulla methods presented here.