ISBN 978-3-8439-3889-1

978-3-8439-3889-1, Reihe Mathematik

Laura Müller
Investigation of moment models for population balance equations and radiative transfer equations

332 Seiten, Dissertation Technische Universität Kaiserslautern (2018), Softcover, A5

Zusammenfassung / Abstract

This thesis deals with moment approximations for population balance and radiative transfer equations. For the radiative transfer equation a particular process is considered, namely the laser induced thermotherapy of liver tumours. Moments are constructed by averaging over a variable against basis functions to produce spectral approximations.

For the first application, the population balance equation, the minimum entropy closure, the polynomial closure and the Kershaw closure are considered. The polynomial closure is a pure spectral method, for which realizability of the moments cannot be guaranteed. That is, the underlying distribution function remains non-negative. The minimum entropy method which is based on a non-negative ansatz overcomes this problem, for certain physically relevant entropies. But this comes at the price of much higher computational costs, since the distribution function is reconstructed by an optimization algorithm. Another closure, able to preserve realizability, is the Kershaw closure. The ansatz consists of a linear combination of Dirac-delta distributions, constructed with the help of information about the realizability boundaries. Additionally, the corresponding partial moment closures will be derived.

For the second application the full and partial linear moment models will be considered in addition to the diffusion approximation. Since these methods could lead to not good enough approximations, the iterative solution scheme to solve the steady state radiative transfer equation, discretized by the discrete ordinate method, is accelerated by the use of a developed multigrid scheme which is based on the Lebedev quadrature rules. Another goal for the application of the thermotherapy is to consider a physically more relevant model. Therefore, the incompressible Navier-Stokes equation is used to model blood and cooling fluid flow and since the domain of interest is heterogeneous for both, the heat and radiative transfer equation, interface conditions need to be taken into account.

To solve the space and time dependency of the moment equations with an efficient method, the Runge-Kutta discontinuous Galerkin scheme will be applied for both applications. Since, the minimum entropy and Kershaw closures fail to work outside the set of realizable moments, for these moment methods techniques are derived such that realizability will be preserved when solved with higher order schemes.