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Abstract

The acceleration and propagation of charged and energetic particles in tenuous and non-relativistically moving astrophysical plasmas is modelled with a variant of the Vlasov–Fokker–Planck equation. The distribution function of the particles is expanded in Cartesian tensors or spherical harmonics. The expansion leads to a system of partial differential equations (PDE) that determines the expansion coefficients. In this PhD thesis we derive new formulae to convert between the expansion coefficients of the Cartesian and spherical harmonic expansions, irrespective of the expansion order. These formulae are equally valid for the Cartesian and spherical multipole expansions of the electrostatic (or gravitational) potential. Moreover, we present a novel way to derive the system of PDEs that is based on operators that act in the Hilbert space of spherical harmonics and their representation matrices. The system of PDEs gained with the operators is a system of advection-reaction equations, that we numerically solve with the discontinuous Galerkin (dG) method. We test our implementation of the dG method to demonstrate that the numerical algorithm is robust. Applying it to simulate the acceleration of charged particles at a parallel shock wave shows that our implementation is suited to simulate astrophysical applications.