F. Iavernaro

L. Brugnano, F. Iavernaro, C. Magherini

In this paper we define an efficient implementation of Runge-Kutta methods of Radau IIA type, which are commonly used when solving stiff ODE-IVPs problems. The proposed implementation relies on an alternative low-rank formulation of the methods, for which a splitting procedure is easily defined. The linear convergence analysis of this splitting procedure exhibits excellent properties, which are confirmed by its performance on a few numerical tests.

F. Iavernaro, F. Mazzia, M. S. Mukhametzhanov et al.

Multi-derivative one-step methods based upon Euler-Maclaurin integration formulae are considered for the solution of canonical Hamiltonian dynamical systems. Despite the negative result that simplecticity may not be attained by any multi-derivative Runge--Kutta methods, we show that the Euler-MacLaurin method of order p is conjugate-symplectic up to order p+2. This feature entitles them to play a role in the context of geometric integration and, to make their implementation competitive with the existing integrators, we explore the possibility of computing the underlying higher order derivatives with the aid of the Infinity Computer.