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978-3-8439-1936-4, Reihe Mathematik

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

125 Seiten, Dissertation Technische Universität Kaiserslautern (2014), Softcover, A5

Zusammenfassung / Abstract

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.