ISBN 978-3-8439-2815-1

978-3-8439-2815-1, Reihe Mathematik

Jochen Kall
ADER Schemes for Systems of Conservation Laws on Networks

341 Seiten, Dissertation Technische Universität Kaiserslautern (2016), Hardcover, A5

Zusammenfassung / Abstract

This work presents high-order explicit finite-volume schemes for networks of hyperbolic balance laws. The networks considered consist of edges, that are governed by hyperbolic balance laws, and vertices that couple adjacent edges by by algebraic coupling conditions to each other and optionally with ODEs residing in the vertices.

The schemes presented are extensions of the ADER type with Toro-Castro as well as Harten-Enquist-Osher-Chakravarthy solvers as building blocks to solve the generalized Riemann problems at the vertices. Both types are asymptotically well balanced, conserve mass exactly, and revert to their respective classical variant at the one-on-one coupling vertex. We give some results on stability of polynomial reconstruction and a theorem providing a direct connection between solvers of TT and TC type.

A generalized Cauchy-Kowalevsky procedure is introduced, capable of treating arbitrary combinations of source terms without the need for repeated symbolic manipulations, further increasing the flexibility of the schemes presented. A technique to generate smooth initial data on arbitrary networks is presented, which makes the design of numerical test cases easier. We further investigate lumped-parameter-models generated by simplification of sub-networks and present a specialized high order scheme of HEOC type for these kind of vertices.

All the schemes presented in this thesis are submitted to extensive numerical tests to demonstrate high-order accuracy, efficiency, stable shock capturing across coupling vertices and applicability so large-scale problems. Simplification strategies for large networks are presented and tested. To that end the schemes presented in this work are applied to a model of the arterial blood flow in a human being and we compare the results to measurement data and numerical simulations.