Recent advances in the numerical solution of conservative problems

The numerical solution of conservative problems, i.e., problems characterized by the presence of constants of motion, is of great interest in the computational practice. Such problems, indeed, occur in many real-life applications, ranging from the nano-scale of molecular dynamics to the macro-scale of celestial mechanics. Often, they are formulated as Hamiltonian problems. Concerning such problems, recently the energy conserving methods named Hamiltonian Boundary Value Methods (HBVMs) have been introduced. In this paper we review the main facts about HBVMs, as well as the existing connections with other approaches to the problem. A few new directions of investigation will be also outlined. In particular, we will place emphasis on the last contributions to the field of Prof. Donato Trigiante, passed away last year.