ISBN 978-3-86853-768-0

978-3-86853-768-0, Reihe Ingenieurwissenschaften

Martin Quandt
High Order Particle Transport for PIC Simulations of Plasma Flows

109 Seiten, Dissertation Universität Stuttgart (2010), Softcover, A5

Zusammenfassung / Abstract

Numerical simulations of plasma flows based on the Particle in Cell (PIC) technique need a procedure for the integration of Newtons relativistic equation of motion for charged particles. In this work a new explicit single step integration method based on a Taylor series expansion of particles velocity is presented.

Up to now the most often used particle push methods are the enhanced leapfrog scheme by J.P. Boris and the classic Runge-Kutta scheme. The special construction of the explicit Boris leapfrog scheme yields to a very efficient and robust integration, but the scheme is limited to a second order convergence rate. For a high order explicit integration the Runge-Kutta method is the only one and achieves its convergence rates by evaluating Newton’s equation of motion at different interim stages. The calculation of the these stages with the complete PIC cycle is the most expensive part of this scheme. Both methods serve as a reference in this work.

The presented truncated Taylor series expansion applied on Newtons equation of motion for charged particle is the first high order explicit single step integration method. The realization of this expansion up to the desired truncation order yields to higher total derivatives of the relativistic velocity and the inverse Lorentz factor. The dependency of these derivatives in time, space and the relativistic velocity itself leads to a complex differential operator. To compute the higher total derivatives of the relativistic velocity, the hierarchical structure of this procedure is utilized to construct the operators by a rearrangement of previously defined operators. Furthermore the unknown total derivatives of the electromagnetic fields are replaced by the application of simple differentiation rules by the given high order partial derivatives in time and space as well as the mixed derivatives. These higher temporal and spatial derivatives of the electromagnetic fields are a prerequisite of the new integration scheme and have to be calculated by a high order Maxwell solver.

To assess and verify this new integration method the Taylor series expansion was tested on different examples in the non-relativistic case on space and time dependent electromagnetic fields and in the relativistic region where the Lorentz factor with all total derivatives are present. For all examples the experimental order corresponds to the selected formal order and a gain in accuracy and efficiency by an increase of the selected formal order is successfully demonstrated.