Hadron collective effects studies for the LHC and FCC-hh project

High intensity particle beams circulating in accelerators can be perturbed by the electromagnetic field created by the beam itself and the interaction with the surrounding environment. This perturbation can potentially drive the whole beam unstable and conduct to particle losses. The concepts of wakefield and impedance allow to study the electromagnetic interaction of the beam with its surroundings and are the starting point to perform numerical beam stability simulations. A way to study beam stability using an impedance model is to solve the Vlasov equation with a perturbative approach: the unstable modes developing in a beam are associated to the solution of an eigenvalue problem. In this work we will present the formalism used in the simulation code DELPHI (Discrete Expansion over Laguerre Polynomials and Headtail modes) that allows to extract the most unstable eigenmodes and the corresponding eigenvectors, directly comparable with the signal measurable with beam position monitors in the machine. We will present studies performed with DELPHI on the effect of different particle distributions on the tune of the machine, and on the Transverse Mode Coupling Instability (TMCI) threshold for the LHC and FCC-hh. Basic studies with classical impedance models (e.g. resistive wall, broad band impedance) will be presented as well