Implementation of Implicit Runge-Kutta Methods

Abstract

If the dimension of the differential equation y′ = f(x, y) is n, then the s -stage fully implicit Runge-Kutta method (3.1) involves a n · s -dimensional nonlinear system for the unknowns g ₁, ... , g _s . An efficient solution of this system is the main problem in the implementation of an implicit Runge-Kutta method.

These have not been used to any great extent ...

(S.P. Nørsett 1976)

However, the implementation difficulties of these methods have precluded their general use; ...

(J.M. Varah 1979)

Although Runge-Kutta methods present an attractive alternative, especially for stiff problems, ... it is generally believed that they will never be competitive with multistep methods.

(K. Burrage, J.C. Butcher & F.H. Chipman 1980)

Runge-Kutta methods for stiff problems, we are just beginning to explore them ...

(L. Shampine in Aiken 1985)