Structure-preserving numerical integration of evolutionary problems

Raffaele D'Ambrosio

Universita' di Salerno

Thursday 17th December 2015

11:00:00

Aula Seminari "Saleri" VI Piano Dipartimento di Matematica, Politecnico di Milano

It is the purpose of this talk to analyze the behaviour of multi-value numerical methods acting as structure-preserving integrators for the numerical solution of some classes of evolutionary problems, such as stochastic differential equations (SDEs), piecewise smooth dynamical systems, Hamiltonian problems and reaction-diffusion problems. The methodology we aim to follow is unifying, i.e. we propose problem-oriented numerical solvers, able to accurately and efficiently reproduce typical properties and behaviors of the above mentioned problems. Thus, according to this perspective, the methods we consider are adapted to the problem, in contrast with general purpose solvers that do not take into account specific features of the problems. The features we aim to consider are given by exponential mean-square contractivity for SDEs, accurately preserving periodic orbits in piecewise smooth dynamical systems, invariants preservation for Hamitonian problems, efficiently reproducing periodic or oscillatory behaviours in reaction-diffusion problems. The presented results deal with a series of joint works in collaboration with Evelyn Buckwar (Johannes Kepler University of Linz), John C. Butcher (University of Auckland), Luca Dieci and Fabio Difonzo (Georgia Institute of Technology), Ernst Hairer (University of Geneva), Beatrice Paternoster (University of Salerno) contact: christian.vergara@polimi.it