Numerical methods for the simulation of particle motion in electromagnetic fields

In this thesis we study numerical methods for the approximate solution of problems arising in electromagnetism. Its main motivations come from applications to the modelling of high-energy particle accelerators. In this framework, we first compare the efficiency of several numerical methods for the omputation of particle trajectories in the design of a magnetic quadrupole for the High Luminosity - Large Hadron Collider (HL-LHC) project and we analyse the use of a specific vector potential gauge to reduce the computational cost. The results from this first comparison motivate the subsequent investigation of the accuracy of the numerical approximation of the field. We therefore develop a new type of discretization for the reconstruction of the magnetic scalar potential in cylindrical domains and we apply it to the field reconstruction from a realistic measurement process in a Bayesian framework. We compare this method with the reconstruction obtained by a more classical method based on the separation of variables, highlighting the benefits of the new type of discretization and its applicability to the reconstruction process. Motivated by the need of efficient methods for the description of electromagnetic fields, we extend the study to other types of problems for axisymmetric domains, which have a high practical relevance in particle accelerator applications. In this context, we propose the use of a method based on the Fourier basis and IsoGeometric Analysis (IGA) to exploit, on one hand, the computational efficiency that can be achieved thanks to the symmetry of the domain and, on the other, the exact representation of the geometry and the good approximation properties achievable in a IGA framework. Moreover, the proposed method forms a de Rham complex, which is a crucial property that allows to obtain a stable method which produces physically correct approximations. We finally apply the method to the computation of resonant modes of an accelerating TESLA cavity