Adaptive methods for the numerical simulation of transport processes [Elektronische Ressource] / Martin Andreas Käser

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Lehrstuhl fur˜ Numerische Mathematikund Wissenschaftliches RechnenProf. Dr. Folkmar BornemannAdaptive Methods for theNumerical Simulationof Transport ProcessesMartin Andreas K˜aserVollst˜andigerAbdruckdervonderFakult˜atfur˜ MathematikderTechnischenUniversit˜at Munc˜ hen zur Erlangung des akademischen Grades eines Doktorsder Naturwissenschaften genehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr. F. BornemannPrufer˜ der Dissertation:1. Priv.-Doz. Dr. A. Iske2. Univ.-Prof. Dr. H. IgelLudwig-Maximilians-Universit˜at Munc˜ hen3. Prof. E. F. Toro, Ph.D.Univ. Trient / ItalienDieDissertationwurdeam09.10.2003beiderTechnischenUniversit˜atMunc˜ heneingereichtunddurchdieFakult˜atfur˜ Mathematikam16.12.2004angenommen.iAcknowledgmentsFirst of all I want to thank my supervisor Armin Iske, who always had thetime to discuss any question and who taught me how to approach numericalproblems efiectively. I enjoyed working together with him not only becauseof his experience and encouraging support, but also because of his patienceand good sense of humor. Many thanks to J˜orn Behrens, whose knowledgeabout the semi-Lagrangian approach and the flnite element method resultedin very important and helpful hints.I am also deeply grateful to all my colleagues of the department for the com-fortable and productive working atmosphere and the many discussions thathelped to look at problems from a difierent view.

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Publié le 01 janvier 2003
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Lehrstuhl fur˜ Numerische Mathematik
und Wissenschaftliches Rechnen
Prof. Dr. Folkmar Bornemann
Adaptive Methods for the
Numerical Simulation
of Transport Processes
Martin Andreas K˜aser
Vollst˜andigerAbdruckdervonderFakult˜atfur˜ MathematikderTechnischen
Universit˜at Munc˜ hen zur Erlangung des akademischen Grades eines Doktors
der Naturwissenschaften genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr. F. Bornemann
Prufer˜ der Dissertation:
1. Priv.-Doz. Dr. A. Iske
2. Univ.-Prof. Dr. H. Igel
Ludwig-Maximilians-Universit˜at Munc˜ hen
3. Prof. E. F. Toro, Ph.D.
Univ. Trient / Italien
DieDissertationwurdeam09.10.2003beiderTechnischenUniversit˜atMunc˜ hen
eingereichtunddurchdieFakult˜atfur˜ Mathematikam16.12.2004angenommen.i
Acknowledgments
First of all I want to thank my supervisor Armin Iske, who always had the
time to discuss any question and who taught me how to approach numerical
problems efiectively. I enjoyed working together with him not only because
of his experience and encouraging support, but also because of his patience
and good sense of humor. Many thanks to J˜orn Behrens, whose knowledge
about the semi-Lagrangian approach and the flnite element method resulted
in very important and helpful hints.
I am also deeply grateful to all my colleagues of the department for the com-
fortable and productive working atmosphere and the many discussions that
helped to look at problems from a difierent view.
Special thanks to Heiner Igel, who enabled me to keep in close contact with
geophysics through his various seminars and workshops. His interest in new
numerical methods, especially for numerical seismology, always encouraged
metokeepinmindtheapplicabilityande–ciencyofnumericalschemeswith
respect to elastic wave propagation.
In particular, I would like to thank Tito Toro, who inspired me to work
on ADER methods. His invitation to the workshops and conferences in
Barcelona and Cambridge ofiered me the great opportunity to get a better
understanding of a variety of numerical methods for hyperbolic conservation
laws. Furthermore, he gave me the chance to present and discuss my own
results in front of an exclusive audience in Cambridge. Thanks, to Randy
LeVeque, Chi-WangShu, andKnut-AndreasLiefortheirhelpfulandencour-
aging comments during this workshop.
My collaborators within the European project NetAGES (Network for Auto-
mated Geometry Extraction from Seismic), Stine Kjersti Richardsen, Tobias
Werther, Alexander Boukhgueim, and Erik Monsen should also be acknowl-
edged for broadening my view on the difierent topics of their work related to
theNetAGESproject. SpecialthankstotheprojectcoordinatorTrygveRan-
den, Magne Lygren, and Brice Valles, who supported my work very actively
and made my visits at Schlumberger Stavanger Research very productive
and successful, in particular while using their commercial software to com-
pare numerical results.
Ich bedanke mich auch herzlich bei meiner Familie und meiner Freundin,ii
die mich bei so viele Dingen stets tatkr˜aftig unterstutzt˜ haben und deren
Ruc˜ khalt fur˜ mich immer von gro…er Bedeutung war.
Munchen,˜ Oktober 2003
Martin K˜aseriv
Zusammenfassung
Zun˜achstwirdeinneues,gitterfreiesVerfahrenzurnumerischenL˜osungnicht-
linearer Transportgleichungen entwickelt. Das Verfahren kombiniert eine
adaptive semi-Lagrange Methode mit einer lokalen Interpolation unter Ver-
wendung von radialen Basisfunktionen. Die vorgestellte Partikelmethode
verwendet eine Menge beliebig verteilter Punkte, wodurch kein Gitter zur
Diskretisierung ben˜otigt wird. Im weiteren wird dieses semi-Lagrange Ver-
fahren erweitert, so da… fur˜ lineare Advektionsprobleme exakte Massenerhal-
tung erlangt wird. Dazu werden adaptive Voronoi-Diagramme verwendet,
um die Flexibilit˜at einer Partikelmethode beizubehalten. Abschlie…end wird
eine neuartige, gitterbasierte Methode (ADER-Methode) auf adaptive, un-
strukturierte Triangulierungen erweitert, die exakte Massenerhaltung auch
fur˜ nichtlineare Transportprobleme liefert. Insbesondere unterliegt die Kon-
vergenzordnung solcher ADER-Methoden keiner Beschr˜ankung und wird bis
zu Verfahren vierter Ordnung untersucht.
Fur˜ alle behandelten Methoden werden sowohl die Fehlersch˜atzer und die
AdaptionsroutinenzurVerfeinerungoderVergr˜oberungderPartikelbzw. des
Gitters ausfuhrlic˜ h diskutiert, als auch die Leistungsf˜ahigkeit dieser adap-
tiven Verfahren anhand numerischer Beispiele unterstrichen. Insbesondere
wird auf ein Anwendungsbeispiel aus dem Bereich der Reservoir Simula-
tion eingegangen, das vor allem in der Erd˜olindustrie zur Modellierung von
Flussigk˜ eitstransport in Lagerst˜atten von gro…em Interesse ist.v
Abstract
At flrst a new meshfree advection scheme for numerically solving nonlin-
ear transport equations is developed. The scheme, being a combination of
an adaptive semi-Lagrangian method and local radial basis function inter-
polation, is essentially a method of backward characteristics. The proposed
particlemethodworkswithanunstructuredsetofnodes,suchthatnomeshis
necessary for the discretization. Furthermore, this semi-Lagrangian method
is modifled in order to achieve exact mass conservation for linear advection
problems. Voronoi diagrams are used to retain the exibilit y of the particle
method. Finally, a new conservative scheme (ADER-scheme) is extended to
adaptive unstructured triangulations in order to treat nonlinear transport
problems. In particular, ADER schemes can be constructed up to an arbi-
trary high order of convergence and are investigated in detail up to order
four.
For all methods the error estimation and the strategy of the adaption rules
forthereflnementorcoarseningofparticlesormeshesarediscussedindetail.
The good performance of the resulting methods is conflrmed by numerical
examples. In particular, a test case from the oil industry is addressed, which
plays an important role in themodelling of uid o w in petroleum reservoirs.