An Efficient Solution Process for Implicit Runge–Kutta Methods

Described in this paper is an efficient solution procedure for a modified Newton method solution of a ν-stage implicit Runge–Kutta method applied to a set of n first order ordinary differential equations. The procedure requires only order ν n³ scalar multiplications for each time-step plus order ν ² n² scalar multiplications for each iteration-step within the time-step. Also, it only requires space to store order 3n³ variables.