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"In circular high energy particle accelerators, such as the High Luminosity LHC (HL-LHC), the stability of the beam is a crucial aspect and it can be assessed simulating the particle trajectories for a large number of revolutions. Certain elements in the accelerator are critical for the overall beam stability and require a more accurate description of their fields and of their effect on the particles. This research deals with the study of efficient methods to compute the particle trajectories which are suitable for long-term simulation (e.g. symplectic methods) and with the study of methods (e.g. IGA) to accurately describe the fields in which the particles are moving.