Some recent advances in the numerical solution of functional equations

This talk is aimed to provide a structure-preserving approach for the numerical solution of evolutionary problems based on functional equations of various type. Starting from known qualitative behaviors of a selection of problems, the chosen approach introduces their numerical counterpart and, as a consequence, such properties are also numerically retained. More specifically, the talk considers multi-value numerical methods for Hamiltonian problems, adapted methods for wavefronts arising in reaction-diffusion problems, nonlinear stability properties in stochastic differential equations, Volterra integral equations with oscillatory solutions, numerical computation of periodic orbits arising in piecewise smooth dynamical systems. For all these problems, aspects more connected to the analysis of the properties of a candidate numerical scheme will be provided, as well as numerical experiments confirming the theoretical expectations. The presented results regard the joint research with Evelyn Buckwar (Johannes Kepler University of Linz), Angelamaria Cardone (University of Salerno), Dajana Conte (University of Salerno), Luca Dieci (Georgia Institute of Technology), Fabio Difonzo (Georgia Institute of Technology), E. Hairer (University of Geneva), Martina Moccaldi (University of Salerno), Beatrice Paternoster (University of Salerno).