Numerical Schemes for Kinetic Equations with Applications to Fibre Lay-Down and Interacting Particles

The present thesis contains efficient numerical schemes for solving Fokker-Planck or Vlasov equations. The relevance and derivation of these kinetic equations as a statistical description of dynamical systems with or without noise will be explained in a brief overview. Higher order schemes suited for high-dimensional kinetic equations are presented, like a Semi-Lagrangian scheme relying on a Bezier curve approach to deal with interpolation and slope limiting at the same time. Furthermore a finite volume method is introduced, that is capable of dealing with anisotropic diffusion on a spherical grid geometry using barycentric interpolation at the grid vertices. These methods are applied to the problems of fibre lay-down and self-propelled, interacting particles. The kinetic equations associated with these models are solved numerically and the results are compared with the microscopic and macroscopic solutions. For systems of self-propelled, interacting particles, milling solutions and their behaviour under stochastic perturbations are analysed numerically.