Georgia Institute of Technology College of Sciences

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 ordinary and partial differential equations (PDEs), with special emphasys to Hamiltonian problems and reaction-diffusion PDEs. As regards Hamiltonian problems, we provide a rigorous long-term error analyis obtained by means of backward error analysis arguments, leading to sharp estimates for the parasitic solution components and for the error in the Hamiltonian. As regards PDEs, we consider structure-preservation properties in the numerical solution of oscillatory problems based on reaction-diffusion equations, typically modelling oscillatory biological systems, whose solutions oscillate both in space and in time. Special purpose numerical methods able to accurately retain the oscillatory behaviour are presented.