On the Starting Algorithms for Fully Implicit Runge-Kutta Methods

Abstract

This paper is concerned with the behavior of starting algorithms to solve the algebraic equations of stages arising when fully implicit Runge-Kutta methods are applied to stiff initial value problems. The classical Lagrange extrapolation of the internal stages of the preceding step and some variants thereof that do not require any additional cost are analyzed. To study the order of the starting algorithms we consider three different approaches. First we analyze the classical order through the theory of Butcher's series, second we derive the order on the Prothero and Robinson model and finally we study the stiff order for a general class of dissipative problems. A detailed study of the orders of some starting algorithms for Gauss, Radau IA-IIA, Lobatto IIIA-C methods is also carried out. Finally, to compare the most relevant starting algorithms studied here, some numerical experiments on well known nonlinear stiff problems are presented.