An Efficient Solution Process for Implicit Runge–Kutta Methods

Abstract

Described in this paper is an efficient solution procedure for a modified Newton method solution of a $\nu $-stage implicit Runge–Kutta method applied to a set of n first order ordinary differential equations. The procedure requires only order $\nu n^3 $ scalar multiplications for each time-step plus order $\nu ^2 n^2 $ scalar multiplications for each iteration-step within the time-step. Also, it only requires space to store order $3n^3 $ variables.