High order well-balanced finite volume schemes for geophysical flows [Elektronische Ressource] : development and numerical comparisons / vorgelegt von Normann Pankratz

Sujets

• Mathematics

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2007
• Nombre de lectures: 24
• Langue: English
• Poids de l'ouvrage: 8 Mo

Extrait

High-order well-balanced finite-volume
schemes for geophysical flows.
Development and numerical comparisons.
Von der Fakult¨at fur¨ Mathematik, Informatik und
Naturwissenschaften der Rheinisch-Westf¨alischen Technischen
Hochschule Aachen zur Erlangung des akademischen Grades
eines Doktors der Naturwissenschaften genehmigte Dissertation.
vorgelegt von
Diplom-Mathematiker
Normann Pankratz
aus Rheinbach
Berichter: Universitatspro¨ fessor Dr. Sebastian Noelle
Universit¨atsprofessor Dr. Gabriella Puppo
Tag der mundlic¨ hen Prufung¨ : 09.Juli 2007
DieseDissertationistaufdenInternetseitenderHochschulbibliothekonlineverfugbar.¨2
Acknowledgements
ThisthesishasbeensupportedbytheDeutscheForschungsgemeinschaftDFG
through a grant of the Graduiertenkolleg 775, Hierarchie und Symmetrie in
mathematischen Modellen.
First of all I would like to thank my supervisor Professor Sebastian Noelle
for four years of successful collaboration. He enthusiastically received my
newly attained insights and continously encouraged me to take the next step.
His ideas and suggestions have been irreplaceable for the complete work.
Special thanks go to the collaboration partners of the two projects. To the
specialist for High-Order schemes Professor Gabriella Puppo for answering
many many questions, providing her codes and offering great hospitality in
Milan. ToProfessorBjørnGjevikfortheveryexcitinggulf-streamproject,for
sharinghisgeophysicalknowledgeandforinvitingmetoOsloandTrondheim.
And also to my friend Jostein Natvig who filled me with enthusiasm for the
geophysicalproblems. DuringhisthreemonthstayinAachenwehavestarted
an intensive collaboration which still continues.
Moreover, I would like to thank:
Ralf Massjung for sharing varied experiences and for his excellent overview of
hyperbolic problems, furthermore for cycling, jogging and hiking tours.
Siegfried Mu¨ller for helpful suggestions.
Marcus Soemers for three summer-schools, various night-sessions, for his hu-
mour and positivity.
Roland Sch¨afer for always being a competent general lexicon, for discussing
design and conceptional questions.
Wolfram Rosenbaum, Kolja Brix, Thies Frings, Andreas Bollermann and
Markus Probst for very important and fruitful discussions.
Frank Knoben for the enduring computer-administration and unresting sup-
port for parallelisation.
Julia Holtermann for her helpfulness and assistance.
Also I would like to thank the whole “Institut fur¨ Geometrie und praktische
Mathematik” for a very nice and enjoyable time, especially for the memorable
seminar in Hirschegg.
Finally,Iwouldliketothankmyparentsfortheirmultifarioussupportand
of course, Christina Steiner for keeping me on the right track, for encouraging
me, loving me, for being invaluable.3
Abstract
Manygeophysicalflowsaremerelyperturbationsofsomefundamentalequilib-
riumstate. Ifanumerical scheme shall capture such flowsefficiently, itshould
be able to preserve the unperturbed equilibrium state at the discrete level.
In the first part of this thesis we present a class of schemes of any desired
order of accuracy which preserve the lake at rest perfectly. These schemes
should have an impact for studying important classes of lake and ocean flows.
In the introduction, we present some of the key ideas and ingredients of the
subsequent sections. We begin with a review of the shallow water equations
and their equilibrium states, in particular the lake at rest. Then we show an
example of a numerical storm produced by a scheme which is not in discrete
equilibrium. Next we review the key ingredient of several of the recent well-
balanced schemes, and give some related references. We close with a preview
of our new high order well-balanced schemes.
Inthe secondpartwe compare a classical finite-difference anda highorder
finite-volume scheme for barotropic ocean flows. We compare the schemes
with respect to their accuracy, stability, and study various outflow and inflow
boundaryconditions. Weapplytheschemestotheproblemofeddyformation
in shelf slope jets along the Ormen Lange section of the Norwegian shelf.
Our results strongly confirm the development of mesoscale eddies caused by
instability of the flows.4Contents
I Shallow Water Flow and Well-Balanced Schemes 9
1 Geophysical Flows 11
1.1 Model Equation for Fluid-Flows. . . . . . . . . . . . . . . . . . 11
1.2 Shallow Water Equations . . . . . . . . . . . . . . . . . . . . . 12
1.3 Gravity waves and hydrostatic pressure . . . . . . . . . . . . . 13
1.4 Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Numerical Storms . . . . . . . . . . . . . . . . . . . . . 15
2 High Order Well-Balance Schemes in One-Dimension 19
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Review of Second-Order Well-Balancing via Hydrostatic Re-
construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Higher Order Well-Balancing via Extrapolation . . . . . . . . . 25
3 High Order Well-Balanced Scheme in Two Dimensions 29
4 Numerical Flux Functions for Finite Volume Schemes 33
4.1 Shallow water 1D Roe Scheme . . . . . . . . . . . . . . . . . . 34
4.2 Lax Friedrichs Scheme . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 HLL Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.4 Shallow water 2D Roe Scheme . . . . . . . . . . . . . . . . . . 38
5 WENO Reconstruction 41
5.1 How to Compute Reconstruction Polynomials . . . . . . . . . . 43
5.2 Reconstruction in Two Dimensions . . . . . . . . . . . . . . . . 45
6 Numerical Experiments: Perturbations of Water at Rest 49
6.1 Order of Accuracy in 1D . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Perturbation of a Lake at Rest in 1D . . . . . . . . . . . . . . . 50
56 CONTENTS
6.3 Dam-break Over a Rectangular Wall . . . . . . . . . . . . . . . 51
6.4 Well-Balanced Test in Two Dimensions . . . . . . . . . . . . . 57
6.5 Order of Accuracy 2D . . . . . . . . . . . . . . . . . . . . . . . 57
6.6 A Small Perturbation of a Steady-State Lake in 2D . . . . . . . 58
6.7 Tang Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
II Barotropic flows: jets in a rotating frame 65
7 Barotropic Flows 67
7.1 Introduction Part Two . . . . . . . . . . . . . . . . . . . . . . . 67
8 Discretisation 71
8.1 The Finite-Difference Scheme . . . . . . . . . . . . . . . . . . . 71
8.1.1 Full Second-Order Extension of Finite-Difference Scheme 73
8.2 Treatment of Coriolis Force in the Finite-Volume Scheme . . . 74
9 Treatment of boundary conditions 77
9.1 Finite-Difference Boundary Conditions . . . . . . . . . . . . . . 77
9.2 Finite-Volume Boundary Conditions . . . . . . . . . . . . . . . 78
9.2.1 Reflective boundary conditions . . . . . . . . . . . . . . 78
9.2.2 Absorbing Outflow Boundary Condition . . . . . . . . . 78
9.2.3 Free-slip inflow boundary conditions . . . . . . . . . . . 86
9.2.4 Balanced Inflow Boundary Condition . . . . . . . . . . . 88
10 Numerical Experiments for Jets in a Rotating Frame 91
10.1 Order of Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 91
10.2 Large Eddies in a Doubly Periodic Domain. . . . . . . . . . . . 92
10.3 Convergence Test for a Barotropic Jet Problem . . . . . . . . . 96
10.4 Development of Eddies in Shelf Slope Area . . . . . . . . . . . 101
10.4.1 Ormen Lange Shelf Experiment I . . . . . . . . . . . . . 101
10.4.2 Setup for Ormen Lange Shelf Experiment II . . . . . . . 103
10.4.3 Setup for Ormen Shelf Experiment III . . . . . . 103
10.4.4 Setup for Ormen Lange Shelf Experiment IV . . . . . . 103
10.4.5 Balancedinflowboundaryconditions: OrmenLangeShelf
Experiment V . . . . . . . . . . . . . . . . . . . . . . . . 107
10.4.6 Comparison with linear stability analysis . . . . . . . . 109
11 Discussion 115
11.1 Efficiency and stability of the FD and FV solvers . . . . . . . . 116
11.2 Numerical inflow boundary conditions . . . . . . . . . . . . . . 116CONTENTS 7
11.3 Geophysical implication of the computational results . . . . . . 117
12 Summary and Future Perspectives 119
Bibliography 120
A Implementation and Parallelisation 1298 CONTENTSPart I
Shallow Water Flow and
Well-Balanced Schemes
9