Dokument: Active Flux Methods for Conservation Laws on Complex Geometries

Titel:	Active Flux Methods for Conservation Laws on Complex Geometries
URL für Lesezeichen:	https://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=57658
URN (NBN):	urn:nbn:de:hbz:061-20211011-083320-6
Kollektion:	Dissertationen
Sprache:	Englisch
Dokumententyp:	Wissenschaftliche Abschlussarbeiten » Dissertation
Medientyp:	Text
Autor:	Dr. Kerkmann, David Christian [Autor]
Dateien:	[Dateien anzeigen] Adobe PDF [Details] 18,10 MB in einer Datei [ZIP-Datei erzeugen] Dateien vom 06.10.2021 / geändert 06.10.2021
Beitragende:	Prof. Dr. Helzel, Christiane [Betreuer/Doktorvater] Prof. Dr. Klein, Rupert [Gutachter]
Stichwörter:	Active Flux, Conservation Laws, Finite Volume Method
Dewey Dezimal-Klassifikation:	500 Naturwissenschaften und Mathematik » 510 Mathematik
Beschreibung:	We are interested in solving hyperbolic conservation laws that are models for transport phenomena for example in computational fluid dynamics. In practical applications, the domain of interest is typically not shaped simply but is rather complex. To discretize these complex geometries, we use a Cartesian grid which cells are cut along the boundary. The resulting cut cells can have arbitrary shapes and sizes. A numerical method for conservation laws should be accurate, stable and conservative. Many methods have been proposed that deal with these aspects for cut cells. Each method has its own advantages and disadvantages. Explicit finite volume methods typically satisfy a time step restriction that depends on the smallest cell size. We search for a third order accurate method that is both conservative and stable for reasonable time steps that do not depend on the size of the smallest cut cells. A necessary requirement for stability is the so-called cancellation property which ensures that the update of the small cells is bounded by the order of their cell sizes. An attractive candidate for a method that might satisfy these properties is the Active Flux method developed by Eymann and Roe. The Active Flux method is a finite volume method that not only uses the cell average values but also point values of the conserved quantities at the interfaces between neighboring cells. It updates the point values separately from the cell averages, thus the flux is computed actively. In each cell, the conserved quantities are reconstructed locally using only the values that belong to that cell. When reconstructing on irregular grid cells, reconstructions can become ill conditioned and lead to poor results when some values are disturbed slightly. We discuss ways to overcome this problem. Furthermore, we examine the stability properties of the Active Flux method for linear systems on Cartesian grids and cut cell grids. The method is stable on cut cell grids for time steps that are restricted only by the size of a regular grid cell. We show that the cancellation property can be achieved when using a continuous reconstruction. No further stabilization technique is required. Furthermore, we find that for our linear model problems the method shows an excellent third order accuracy and is conservative by nature. In the second part we develop a different interpretation of the Active Flux method for Cartesian grids. This reinterpretation can be applied to every hyperbolic conservation law and is no longer restricted to linear or simple nonlinear systems. It differs from the original Active Flux method in the way the point values are updated since no exact evolution formula is known. We present results for a multitude of test cases and obtain third order accuracy, good stability and conservation. Many of the results presented in this thesis can be found in our previous publications.
Lizenz:	Urheberrechtsschutz
Fachbereich / Einrichtung:	Mathematisch- Naturwissenschaftliche Fakultät » WE Mathematik » Angewandte Mathematik
Dokument erstellt am:	11.10.2021
Dateien geändert am:	11.10.2021
Promotionsantrag am:	24.06.2021
Datum der Promotion:	30.09.2021