Recent advances in numerical modeling for differential problems

It is the purpose of this talk to analyze the behaviour of some classes of numerical methods acting as structure-preserving integrators for the numerical solution of ordinary (ODEs) and partial differential equations (PDEs), with special emphasys to Hamiltonian problems, discontinuous ODEs, reaction-diffusion problems and stochastic differential equations (SDEs). 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. As regards ODEs, we provide a rigorous long-term error analysis in energy conservation for Hamiltonian problems by multi-value methods, obtained by means of backward error analysis arguments, and provide a study of the stability of periodic orbits in discontinuous ODEs by analyzing the behaviour of the monodromy matrix and Floquet multipliers associated the the orbit. Concerning PDEs, we present an adapted method of lines for problems with periodic or oscillatory solutions, by means of non-polynomially fitted finite differences. The coefficients of the resulting methods will depend on the unknown values of parameters related to the problem (e.g. the values of the frequencies of the oscillations): suitable techniques leading to estimates of the parameters will be discussed. Focusing on nonlinear SDEs, we analyze the conditional properties of stochastic linear multistep methods to retain dissipativity properties. Exponential mean square contractivity is analyzed, through results revealing some conditional nonlinear stability properties leading to accurate bounds for the stepsize. Numerical experiments on a selection of nonlinear problems are presented. 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 and Martina Moccaldi (University of Salerno). Part of this work has been granted by the Fulbright project "Discontinuous Dynamical Systems: An Accurate and Efficient Framework for Their Numerical Solution".