Dissertation Andreas Hiltebrand

171 Pages·2015·11.32 MB·English

by  Andreas Hiltebrand

ETH Library
Entropy-stable discontinuous
Galerkin finite element methods
with streamline diffusion and
shock-capturing for hyperbolic
systems of conservation laws
Doctoral Thesis
Author(s):
Hiltebrand, Andreas
Publication date:
2014
Permanent link:
https://doi.org/10.3929/ethz-a-010337672
Rights / license:
In Copyright - Non-Commercial Use Permitted
This page was generated automatically upon download from the ETH Zurich Research Collection.
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DISS. ETH NO. 22279
ENTROPY-STABLE DISCONTINUOUS GALERKIN
FINITE ELEMENT METHODS
WITH STREAMLINE DIFFUSION AND SHOCK-CAPTURING
FOR HYPERBOLIC SYSTEMS OF CONSERVATION LAWS
A thesis submitted to attain the degree of
DOCTOR OF SCIENCES of ETH ZURICH
(Dr. sc. ETH Zurich)
presented by
ANDREAS EDUARD HILTEBRAND
MSc ETH CSE, ETH Zurich
born on 17.03.1987
citizen of Bachenbu¨lach ZH, Switzerland
accepted on the recommendation of
Prof. Dr. Siddhartha Mishra, ETH Zurich, examiner
Prof. Dr. R´emi Abgrall, University of Zurich, co-examiner
Prof. Dr. Charalambos Makridakis, University of Sussex, co-examiner
2014
ISBN 978-3-906031-85-9
DOI 10.3929/ethz-a-010337672
Abstract
We propose and analyse a space-time discontinuous Galerkin (DG) finite element method
for hyperbolic conservation laws. Entropy stability is obtained by discretising the entropy
variablesandusingentropy-stablenumericalfluxes. Assolutionsofhyperbolicconservation
lawscandevelopdiscontinuities(shocks)infinitetime,weincludeastreamlinediffusionand
a shock-capturing term to suppress spurious oscillations in the vicinity of shocks. We show
that the approximate solutions converge to an entropy measure-valued solution for systems
of conservation laws.
Algorithmically, the discretisation leads to a big nonlinear system of algebraic equations
in each time step. A Newton-Krylov method is applied to solve it. However, for effi-
ciencyreasons, apreconditionermustbeused. WedesignandapplyblockJacobiandblock
Gauss-Seidel type preconditioners and investigate their performance both analytically (in a
simplified setting) and experimentally.
The space-time formulation allows an easy local adaptation of the mesh in space and
time. Weconsiderresidual-basedadaptivityaswellasgoal-orientedadaptivityusingduality
estimates.
Due to the implicitness of the scheme, there is no CFL condition that must be satisfied.
Therefore, the scheme can also be used in problems with multiple time scales, such as flows
near the incompressible limit.
The high resolution and robustness properties of the method are demonstrated in several
experiments. We consider, among others, the linear advection equation, Burgers’ equation,
thewaveequation,andtheEulerequationsinoneortwospatialdimensions. Theseexamples
show that also problems with complicated boundaries can be solved, thanks to the finite
element formulation.
iii

Zusammenfassung
Wir entwickeln und analysieren eine unstetige Galerkin-Finite-Element-Methode in Raum-
zeit fu¨r hyperbolische Erhaltungsgleichungen. Entropie-Stabilit¨at erlangen wir durch das
DiskretisierenderEntropievariablenunddasVerwendenEntropie-stabilernumerischerFlu¨s-
se. Da L¨osungen hyperbolischer Differentialgleichungen in endlicher Zeit Unstetigkeiten
(St¨osse) entwickeln k¨onnen, binden wir einen Stromliniendiffusions- und einen Stossaufl¨o-
sungstermmitein,umst¨orendeOszillationenbeiSt¨ossenzuunterdru¨cken.Wirzeigen,dass
dieapproximativenL¨osungenfu¨rSystemehyperbolischerDifferentialgleichungengegeneine
masswertige Entropiel¨osung konvergieren.
Aus algorithmischer Sicht fu¨hrt die Diskretisierung in jedem Zeitschritt auf ein grosses
nichtlineares System algebraischer Gleichungen. Um es zu l¨osen, benutzen wir eine Newton-
Krylov-Methode. Aus Effizienzgru¨nden muss jedoch ein Vorkonditionierer angewandt wer-
den. Wir entwerfen und verwenden Vorkonditionierer vom Block-Jacobi- und Block-Gauss-
Seidel-TypunduntersuchenihreEffizienzsowohlanalytisch(untervereinfachtenBedingun-
gen) als auch experimentell.
Die Raumzeit-Formulierung erlaubt eine einfache lokale Adaptition des Gitters in Raum
und Zeit. Wir betrachten Adaptivit¨at basierend auf dem Residuum sowie Ziel-orientierte
Adaptivit¨at unter Verwendung von Dualit¨atsabsch¨atzungen.
DankderImplizit¨atdesVerfahrensmusskeineCFL-Bedingungerfu¨lltwerden.Daherkann
dieMethodeauchfu¨rMulti-Zeitskalen-Problemegebrauchtwerden,wiez.B.fu¨rStr¨omungen
nahe dem inkompressiblen Grenzfall.
Die hohe Au߬osung und Robustheit des Verfahrens werden in diversen Experimenten
demonstriert. Wir betrachten unter anderem die lineare Advektionsgleichung, die Burgers-
gleichung,dieWellengleichungunddieEulergleichungineineroderzweir¨aumlichenDimen-
sionen. Zus¨atzlich zeigen diese Beispiele auch, dass Probleme mit komplizierten R¨andern
dank der Finite-Element-Formulierung gel¨ost werden k¨onnen.
v

Acknowledgements1
I would like to thank all the people that have made this thesis possible either directly or
indirectly. First of all, I would like to thank Siddhartha Mishra for all the inspiring ideas,
for the insights into the theory given to me, and for all the time spend in discussions and in
assisting me in any respect.
Furthermore, I would like to mention and thank the following people explicitly: Blanca
AyusodeDiosforthehelpinpreconditioningandforpointingoutthatsuperapproximation
canbeused,FranziskaWeberforherpatienceinseeminglyendlessdiscussionsonprojection
errorsandforenablingmetounderstandcertainaspectsofthetheorymorethoroughly,and
SandraMayforposingapparentlysimplequestionsthathighlightedfundamentalproperties
of the schemes and allow to embed the scheme into the existing work. Moreover, I would
like to thank her, Deep Ray, and Ulrik Fjordholm for critical reading of the thesis, which
has improved it a lot.
I would like to thank all the SAMies that have made the stay at the institute a pleasant
time in my life. This includes the discussions on various topics, being scientific or non-
scientific.
Lastbutnotleast,Iwouldliketothankmyfamily. Withouttheirsupport,whichenabled
me to concentrate my work, this thesis would just not have been possible.
1TheresearchwasfundedinpartbyERCSTG.NN306279SPARCCLE.
vii

Contents
. Abstract iii
. Zusammenfassung v
. Acknowledgements vii
. Introduction xiii
. Notation xvii
1. Hyperbolic conservation laws 1
1.1. Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1. Integral formulation of conservation laws. . . . . . . . . . . . . . . . . 1
1.1.2. Differential formulation of conservation laws . . . . . . . . . . . . . . . 1
1.1.3. Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.4. Weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.5. Entropy conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.6. Entropy measure-valued solutions. . . . . . . . . . . . . . . . . . . . . 4
1.2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1. Scalar conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2. Linear symmetrisable systems . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3. Nonlinear systems of conservation laws. . . . . . . . . . . . . . . . . . 7
2. Numerical approximation of hyperbolic conservation laws 9
2.1. Finite volume schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1. High-resolution schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2. Entropy-stable finite volume schemes . . . . . . . . . . . . . . . . . . . 11
2.2. Discontinuous Galerkin schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3. The shock-capturing streamline diffusion DG formulation 15
3.1. The mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2. Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3. The DG quasilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.1. Numerical fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.2. Entropy-conservative flux . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.3. Numerical diffusion operators . . . . . . . . . . . . . . . . . . . . . . . 18
ix