Variable-order starting algorithms for implicit Runge–Kutta methods on stiff problems

Abstract

This paper deals with starting algorithms for Newton-type schemes for solving the stage equations of implicit s-stages Runge–Kutta methods applied to stiff problems. We present a family of starting algorithms with orders from 0 to s+1 and, with estimations of the error in these algorithms, we give a technique for selecting, at each step, the most convenient in the family. The proposed algorithms, that can be expressed in terms of divided differences, are based on the Lagrange interpolation of the stages of the last two integration steps. We also analyse the orders of the starting algorithms for the non-stiff case, for the Prothero and Robinson model and the stiff order. Finally, by means of some numerical experiments we show that this technique allows, in general, to greatly improve the performance of implicit Runge–Kutta methods on stiff problems.