Line integral methods: An ICNAAM story, and beyond

In recent years, the class of Line Integral Methods has been devised for efficiently solving conservative problems, namely, problems with invariants of motion. Their main instance is given by Hamiltonian Boundary Value Methods (HBVMs), a class of energy-conserving Runge-Kutta methods for Hamiltonian problems. It turns out that the efficient derivation of such methods relies on a local Fourier expansion of the vector field along a suitable orthonormal basis. However, this procedure can be also regarded as a spectral expansion, which can be adapted to a number of differential problems. The foundation of the theory and relevant paths of investigation underlying this class of methods have been developed during the past editions of the ICNAAM Conference series.