Multivalue numerical methods for partitioned differential problems: from second order ODEs to separable Hamiltonians

This talk introduces general families of multivalue methods for the numerical solution of partitioned differential problems which includes, as special cases, second order ODEs of special type and Hamiltonian problems with separable Hamiltonian. We present the family of General Nystrom methods, introduced in [2] with the initial aim to provide an unifying approach for the analysis of minimal accuracy and stability demandings. Within this family, we derive high order P-stable formulae which result to be competitive with respect to classical Runge-Kutta-Nystrom methods in numerically solving periodic sti problems [3]. As regards problem (2), the family of partitioned general linear methods is introduced [1]. With the aim of obtaining a long-term near conservation of the Hamiltonian of (2), we present suitable notions of G-symplecticity and symmetry for these methods. A technique for the generation of numerical solutions with bounded parasitic components is also discussed. Numerical experiments on a selection of separable Hamiltonian problems are reported. The talk deals with a series of joint papers with John Butcher (University of Auckland) and Beatrice Paternoster (University of Salerno). References 1. J. C. Butcher, R. D'Ambrosio, Partitioned general linear methods for separable Hamiltonian problems, in preparation. 2. R. D'Ambrosio, E. Esposito, B. Paternoster, General linear methods for y'' = f(y(t)), Numer. Algor. 61(2), 331{349 (2012). 3. R. D'Ambrosio, B. Paternoster, P-stable General Linear Nystrom methods, submitted.