Multidimensional systems of hyperbolic conservation laws, numerical schemes, and characteristic theory [Elektronische Ressource] : connections, differences, and numerical comparison / vorgelegt von Tim Kröger

Sujets

• Mathematics

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2004
• Nombre de lectures: 12
• Langue: English
• Poids de l'ouvrage: 15 Mo

Extrait

Multidimensional systems of
hyperbolic conservation laws,
numerical schemes,
and characteristic theory
|
Connections, di erences,
and numerical comparison
Tim Kroger.Multidimensional systems of
hyperbolic conservation laws,
numerical schemes,
and characteristic theory
|
Connections, di erences,
and numerical comparison
Von der Fakultat fur Mathematik, Informatik und
Naturwissenschaften der Rheinisch-Westfalischen
Technischen Hochschule Aachen zur Erlagung des
akademischen Grades eines Doktors der
Naturwissenschaften genehmigte Dissertation
vorgelegt von
Diplom-Mathematiker Tim Kroger
aus Hamburg
Berichter:
Universitatsprofessor Dr. Sebastian Noelle
Universitatsprofessorin Dr. Maria Luk acov a
Tag der mundlichen Prufung: 16. Juli 2004
Diese Dissertation ist auf den Internetseiten
der Hochschulbibliothek online verfugbar..Acknowledgements
The present dissertation was partially funded by the DFG-priority
research program ‘Analysis and Numerics for Conserva-
tion Laws’ (ANumE). I would like to thank the Deutsche For-
schungsgemeinschaft for this kind of support. The remaining part of
my position was nanced by the ‘Institut fur Geometrie und
Praktische Mathematik’ (IGPM) of the RWTH Aachen, who
also provided the necessary infrastructure for the preparation of this
thesis. I would like to thank them as well. Also, I would like to
thank my supervisor Professor Sebastian Noelle for the great
support, many fruitful and critical discussions, and for also sharing
my kind of humor.
Furthermore, I would like to thank
Professor Maria Lukacova for serving as a co-referee for
my thesis and for inviting me for a three month research coop-
eration to Hamburg-Harburg Technical University;
Ralf Massjung for many funny and fruitful discussions near
the electric kettle and especially for the term ‘spinal cord prob-
lem’ for the test problem examined in Subsection 5.4.4;
Olaf Lotter for frequent telephone calls (most of them free
Aof charge for both of us) and for recommending me the LT XE
package ushort;
Isolde Theves, who shared an instructive, salutary and pleas-
ant friendship with me over some period;
my o ce mates Wolfram Rosenbaum and Frank Schmitt
for the wonderful community;
Matthias Wesenberg and Friedemann Kemm for discus-
sions about the MHD equations;
vDonald E. Knuth for the development of T X, Leslie Lam-E
Aport for the development of LT X, Markus Kohm for theE
development of the KoMa document classes, and Norbert
Schwarz for writing his helpful book [48] (and again the IGPM
for having this book in their local library); and
everyone I have forgotten.
Finally, I thank anyone who did things that made me laugh, espe-
cially the ‘Aachener Stra enbahn und Energieversorgungs-
Aktiengesellschaft’ (ASEAG), who claimed that painting one
of their buses with penguins o ers people a more comfortable and
environmentally friendly way to get to the zoo.
viContents
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overall notation . . . . . . . . . . . . . . . . . . . . . 4
2 Systems of hyperbolic conservation laws 7
2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The wave equation system . . . . . . . . . . . . . . . . 9
2.3 The Euler equations of gas dynamics . . . . . . . . . . 11
2.4 The equations of ideal magnetohydrodynamics (MHD) 13
2.4.1 Case A (the generic case) . . . . . . . . . . . . 19
2.4.2 Case B (Bp = 0) . . . . . . . . . . . . . . . . 20
22.4.3 Case C-3 (Bp = 0 andjBj <p) . . . . . . 22
22.4.4 Case C-1 (Bp = 0 andjBj >p) . . . . . . 24
22.4.5 Case C-2 (Bp = 0 andjBj =p) . . . . . . 24
2.4.6 Case D (Euler case: B = 0) . . . . . . . . . . . 26
2.4.7 The situation for the classical form of the MHD
equations . . . . . . . . . . . . . . . . . . . . . 28
3 A short course on multi-dimensional characteristic theory 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Characteristic curves of a scalar, rst-order equation . 32
3.3 surfaces for a system of rst-order equa-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Bicharacteristic curves . . . . . . . . . . . . . . . . . . 42
3.5 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 Application to the wave equation system . . . . . . . . 51
3.7 to the Euler system . . . . . . . . . . . . . 52
3.8 Application to the MHD . . . . . . . . . . . . 53
viiContents
4 New derivation of the MoT-ICE and connections to other
numerical approaches 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Decompositions of hyperbolic systems . . . . . . . . . 68
4.3 Integral representations for smooth solutions and their
approximation . . . . . . . . . . . . . . . . . . . . . . 71
4.3.1 Integral representation . . . . . . . . . . . . . . 72
4.3.2 Approximate evolution operators . . . . . . . . 74
4.4 A state decomposition based on characteristic theory . 78
4.5 Euler’s equations for varying space dimension . . . . . 84
4.6 Flux decompositions based on gas-kinetic theory . . . 90
4.6.1 Euler’s equations . . . . . . . . . . . . . . . . . 90
4.6.2 Boltzmann’s equation . . . . . . . . . . . . . . 91
4.6.3 Kinetic schemes . . . . . . . . . . . . . . . . . . 92
4.6.4 Derivation of ux decompositions . . . . . . . . 94
4.7 Comparison with Ostkamp’s integral representation . . 105
4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 111
5 Numerical results and comparison of the Method of Trans-
port to a standard scheme 115
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Description of the HLL scheme . . . . . . . . . . . . . 117
5.3 of the MoT-ICE scheme . . . . . . . . . . 120
5.3.1 Prediction step . . . . . . . . . . . . . . . . . . 120
5.3.2 Instabilities at CFL numbers near unity . . . . 122
5.3.3 Numerical approximation of the coupling terms 125
5.3.4 Reconstruction . . . . . . . . . . . . . . . . . . 137
5.4 MoT versus HLL . . . . . . . . . . . . . . . . . . . . . 141
5.4.1 Static contact . . . . . . . . . . . . . . . . . . . 141
5.4.2 disc problem . . . . . . . . . . . . . . . . 148
5.4.3 Sod-2D problem . . . . . . . . . . . . . . . . . 148
5.4.4 Periodic oblique advection problem . . . . . . . 160
5.4.5 Rotor problem . . . . . . . . . . . . . . . . . . 163
5.4.6 Shu{Osher problem . . . . . . . . . . . . . . . 169
5.4.7 Oblique shear wave . . . . . . . . . . . . . . . . 171
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 180
6 Final remarks 183
viiiContents
Bibliography 185
Index 191
Lebenslauf 201
ix.