On the geometric structure, well-posedness and systematic discretization of linear kinetic equations
Vincent Bosboom is a PhD Student in the department Mathematics of Computational Science. (Co)Promotors are prof.dr.ir. J.J.W. van der Vegt and dr. M. Schlottboom from the faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente.
In this thesis, we study the structure of the polarized radiative transfer equation (RTE) and provide an analysis and a novel discretization strategy for the Fokker-Planck equation. As an introduction, in Chapter 1 we provide a phenomenological derivation of the radiative transfer equation, based on Fermat's principle and the properties of electromagnetic waves. This derivation highlights an important property of the RTE, namely that the Stokes parameters Q and U are not physical quantities, but depend on a chosen frame of reference along the light ray. Mathematically, this implies that the solution of the RTE depends on the choice of the Darboux frame (T,t,u) and the resulting functions n and T related to the optical rotation and scattering. Additionally, we introduce the Fokker-Planck equation, by relating it to the RTE with inelastic scattering as an asymptotic limit in the continuous slowing-down approximation. We use this equation to model the propagation of electrons in a magnetic field by including the influence of the Lorentz force.
Chapter 2 is devoted to the analysis of the well-posedness of the radiative transfer equation in Lp spaces on both bounded and unbounded domains. We prove well-posedness on unbounded domains using techniques from semigroup theory. To apply these techniques, we show that the transport term in the RTE generates a strongly continuous semigroup, and that all other terms are bounded. To show the latter, specifically for the scattering terms, we choose suitable norms and develop a novel inequality for the traces of matrices. To show well-posedness on bounded domains we develop novel trace theorems and an integral transformation on the characteristic curve of the flow map, which generalize previous results for the specific case where the characteristic curves are straight lines. Besides well-posedness we also show physical properties of the solution, namely that it remains positive definite and Hermitian.
In Chapter 3 we focus on the geometric structure of the RTE and construct a metriplectic formulation for it. We construct this formulation by considering the time evolution of functionals of the solution and showing that the evolution of these functionals is governed by a Poisson bracket, a metric bracket and a free energy functional. The main challenge in constructing this formulation is finding a suitable Poisson bracket that satisfies the Jacobi identity. We construct such a bracket by a change of reference frame such that the rotation function n vanishes. Finding a proper Poisson bracket for general reference frames with a non-vanishing n still remains an open problem.
In Chapter 4 we study the Fokker-Planck equation describing the propagation of charged particles in the presence of a magnetic field. In this chapter, we first prove the existence of weak solutions to this equation using a Rothe method, which requires the development of a new trace estimate for the proper function spaces. Secondly, we propose a discretization scheme using finite elements in space, a truncated spherical harmonics expansion in angle, and an implicit Euler scheme in energy. We provide a full error analysis of this scheme and apply it to several numerical examples.
