Recent developments with implicit Runge-Kutta algorithms

Implicit Runge-Kutta algorithms, in contrast to the explicit algorithms, are easily derivable to any order. The primary disadvantage of the method is the often slow convergence of the interative procedures that are inherent to the implicit algorithms. Recent developments have alleviated this disadvantage. Furthermore, efficient algorithms have been developed that include a reliable automatic step size control. The coefficients for implicit Runge-Kutta algorithms are derived for systems of first and of second order ordinary differential equations. The strategy for the acceleration of the convergence of a step is developed, and the stepsize control and error control are developed. Numerical comparisons are made for a selection of test problems between the implicit Runge-Kutta methods and explicit Runge-Kutta, multistep, and extrapolation methods.