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- Herausgeber: GRIN Verlag
- Sprache: Englisch
Doctoral Thesis / Dissertation from the year 2007 in the subject Mathematics - Analysis, University of Constance (Fachbereich Mathematik & Statistik), language: English, abstract: Lattice-Boltzmann algorithms represent a quite novel class of numerical schemes, which are used to solve evolutionary partial differential equations (PDEs). In contrast to other methods (FEM,FVM), lattice-Boltzmann methods rely on a mesoscopic approach. The idea consists in setting up an artificial, grid-based particle dynamics, which is chosen such that appropriate averages provide approximate solutions of a certain PDE, typically in the area of fluid dynamics. As lattice-Boltzmann schemes are closely related to finite velocity Boltzmann equations being singularly perturbed by special scalings, their consistency is not obvious. This work is concerned with the analysis of lattice-Boltzmann methods also focusing certain numeric phenomena like initial layers, multiple time scales and boundary layers. As major analytic tool, regular (Hilbert) expansions are employed to establish consistency. Exemplarily, two and three population algorithms are studied in one space dimension, mostly discretizing the advection-diffusion equation. It is shown how these model schemes can be derived from two-dimensional schemes in the case of special symmetries. The analysis of the schemes is preceded by an examination of the singular limit being characteristic of the corresponding scaled finite velocity Boltzmann equations. Convergence proofs are obtained using a Fourier series approach and alternatively a general regular expansion combined with an energy estimate. The appearance of initial layers is investigated by multiscale and irregular expansions. Among others, a hierarchy of equations is found which gives insight into the internal coupling of the initial layer and the regular part of the solution. Next, the consistency of the model algorithms is considered followed by a discussion of stability. Apart from proving stability for several cases entailing convergence as byproduct, the spectrum of the evolution operator is examined. Based on this, it is shown that the CFL-condition is necessary and sufficient for stability in the case of a two population algorithm discretizing the advection equation. Furthermore, the presentation touches upon the question whether reliable stability statements can be obtained by rather formal arguments. To gather experience and prepare future work, numeric boundary layers are analyzed in the context of a finite difference discretization for the one-dimensional Poisson equation.
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Veröffentlichungsjahr: 2007
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Sanctifica mihi omne primogenitum.
Primitias frugum terrae tuae deferes in domum Domini Dei tui.
Vulgata, liber exodi (13,2 & 23,19)
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Preface - Vorwort
Der Anstoß zu meinem Dissertationsvorhaben entstammt dem BMBF-Projekt “Adaptive Gittersteuerung f¨ ur Lattice-Boltzmann Verfahren zur Simulation von F¨ ullprozessen im Gießereibereich” amFraunhofer Institut f¨ ur Techno- und Wirtschaftsmathematik (ITWM)in Kaiserslautern. Im Laufe der Projektarbeit ergaben sich grundlegende Fragen, von denen einige Gegenstand der vorliegenden Dissertationsschrift geworden sind, welche einen Großteil meiner Arbeit der Jahre 2003 bis 2006 zusammenfaßt.
Mein besonderer Dank ergeht an Herrn Prof. Dr. Michael Junk f¨ ur seine wertvolle und wohlwollende Unterst¨ utzung. Sein best¨ andiges Interesse sowie die Freiheit, welche er mir gew¨ ahrt hat, um eigenen Problemstellungen nachzugehen, haben diese Arbeit entscheidend miterm¨ oglicht. Gerne werde ich mich an die vielen Gespr¨ ache mit ihm w¨ ahrend meiner Assistentenzeit an der Universit¨ at Konstanz zur¨ uckerinnern: sei es im Allgemeinen, z.B. bei der Planung interessanter Lehrveranstaltun-gen, oder im Speziellen, wenn der Motor mal wieder ein Tr¨ opfchen ¨ Ol ben¨ otigte, um nicht zu stottern.
Desweiteren danke ich Herrn Prof. Dr. Axel Klar f¨ ur die ¨ außerst z¨ ugige ¨ Ubernahme
des Koreferats, ohne die der enge Zeitplan des “Endspurts” nicht h¨ atte eingehalten werden k¨ onnen.
Mit dem Vollzug der Promotion endet ein Lebensabschnitt, der f¨ ur mich in Kaiserslautern begann und ¨ uber ein kurzes Intermezzo in Saarbr¨ ucken nach Konstanz an den Bodensee f¨ uhrte. Den vielen netten Menschen, die mich auf diesem Wege fachlich oder privat begleitet haben, m¨ ochte ich ebenfalls meinen Dank aussprechen. Dabei denke ich an etliche Mitarbeiter und Mitdoktoranden vom ITWM (insbesondere aus den Abteilungen SKS und HPC). Außerdem sind vor allem meine Freunde Bernd B¨ uchler, Joshua Karnes und Michael Hawlitzky sowie mein Bruder Peter zu nennen. Die gemeinsamen, teils abenteuerlichen Unternehmungen erscheinen wie Inseln im Meer eines arbeitsreichen Alltags.
Den Pastoren Dietrich Schindler und Jeff Ingram danke ich f¨ ur die herzliche Aufnahme in dieFreie Evangelische Gemeinde (FEG)Kaiserslautern. Von den vielen lieben Bekannten dort sei insbesondere Familie Markutzik nebst Hulda erw¨ ahntdie zahlreichen gem¨ utlichen Diskussionsabende sind unvergessen.
Aufgrund meiner Lehraufgaben im Fachbereich Mathematik und Statistik der Universit¨ at Konstanz war die Arbeit seit Oktober 2004 sehr intensiv. F¨ ur die angenehme Arbeitsatmosph¨ are sei dem Fachbereich und insbesondere den Kollegen der Arbeitsgruppe Numerik gedankt. Unsere konversationsreiche Mittagstischrunde habe ich des ¨ ofteren wie das “Luftholen beim Streckentauchen” empfunden.
Nicht zuletzt bin ich meiner Verlobten und meinen Eltern zu tiefem Dank verpflichtet; Vita f¨ ur ihre Geduld aus der N¨ ahe, Mutter und Vater f¨ ur die Einf¨ uhlsamkeit aus der Ferne und allen dreien f¨ ur die notwendige moralische Ermutigung.
Konstanz, den 30. Juli 2007 Martin K. Rheinl¨ ander
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Lattice-Boltzmann algorithms represent a quite novel class of numerical schemes, which are used to solve evolutionary partial differential equations (PDE). In contrast to finite difference and finite element schemes, lattice-Boltzmann methods rely on a mesoscopic (kinetic) approach. The essential idea consists in setting up an artificial, grid-based particle dynamics, which is chosen such that appropriate averages provide approximate solutions of a certain PDE, typically in the area of fluid dynamics. As lattice-Boltzmann schemes are closely related to finite velocity Boltzmann equations being singularly perturbed by special scalings, their consistency is not obvious, however.
This work is concerned with the analysis of lattice-Boltzmann methods, where a particular interest lies in some numeric phenomena like initial layers, multiple time scales and boundary layers. As major analytic tool, regular expansions (Hilbert expansion) are employed to establish consistency. Exemplarily, two and three population algorithms are studied in one space dimension, mostly discretizing the advection-diffusion equation. It is shown how these ‘model schemes’ can be derived from two dimensional schemes in the case of special symmetries. The analysis of the schemes is preceded by an examination of the singular limit being characteristic of the corresponding scaled finite velocity Boltzmann equations. Convergence proofs are obtained using a Fourier series approach and alternatively a general regular expansion combined with an energy estimate. The appearance of initial layers is investigated by multiscale and irregular expansions. Among others, a hierarchy of equations is found which gives insight into the internal coupling of the initial layer and the regular part of the solution.
Next, the consistency of the model algorithms is considered followed by a discussion of stability. Apart from proving stability for several cases entailing convergence as byproduct, the spectrum of the evolution operator is examined in detail. Based on this, it is shown that the CFL-condition is necessary and sufficient for stability in the case of a two population algorithm discretizing the advection equation. Furthermore, the presentation touches upon the question whether reliable stability statements can be obtained by rather formal arguments. To gather experience and prepare future work, numeric boundary layers are analyzed in the context of a finite difference discretization for the one-dimensional Poisson equation.
Sommaire
Les m´ ethodes de Boltzmann sur reseau r´ epresentent une classe de sch´ emas num´ eriques assez nouveaux, qui ont ´ et´ e d´ evelopp´ es pour la r´ esolution des ´ equations aux d´ eriv´ ees partielles (EDP) du type ´ evolutionnaire. Contrairement aux diff´ erences finies et aux ´ el´ ements finis, les m´ ethodes de Boltzmann sur r´ eseau s’inspirent d’une approche m´ esoscopique (cin´ etique). L’id´ ee fondamentale consiste ` a ´ etablir une dynamique artificielle pour des particules imaginaires mouvant sur les liens d’un r´ eseau. Les param` etres de cette dynamique peuvent ˆ etre choisis de telle sorte que des quantit´ es moyenn´ ees convergent vers la solution d’une EDP dont l’origine est
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notamment la m´ ecanique des fluides. Toutefois, la consistance des sch´ emas de Boltzmann sur r´ eseau n’est pas ´ evidente, puisqu’ils sont ´ etroitement associ´ es aux ´ equations de Boltzmann aux v´ elocit´ es finies devenant singuli` erement perturb´ ees si le nombre de Knudsen est petit.
Cet ouvrage est consacr´ e ` a l’analyse des m´ ethodes de Boltzmann sur r´ eseau. Un int´ erˆ et particulier est accord´ e aux ph´ enom` enes num´ eriques comme les couches initiales, les multi-´ echelles temporaires et les couches de bord. Des expansions r´ eguli` eres (expansions de Hilbert) sont utilis´ ees comme outil principal pour v´ erifier
la consistance. ` A titre d’exemple, des algorithmes ` a deux ou trois populations en dimension une de l’espace sont ´ etudi´ es. D’abord, ces algorithmes de mod` ele - surtout ceux r´ esolvant l’´ equation d’advection-diffusion - sont d´ eriv´ es comme cas sp´ eciaux des algorithmes bidimensionnels en exploitant certaines sym´ etries.
L’analyse des sch´ emas est pr´ ec´ ed´ ee par l’examen de la limite singuli` ere ´ etant caract´ eristique des ´ equations de Boltzmann aux v´ elocit´ es finies. En employant des s´ eries de Fourier ou alternativement une expansion r´ eguli` ere g´ en´ erale en combinaison avec une majoration de l’´ energie, des preuves de convergence sont donn´ es. L’apparition des couches initiales est ´ etudi´ ee ` a l’aide d’expansions irr´ eguli` eres et multi-´ echelles. Entre autres, on trouve une hi´ erarchie des ´ equations qui donne des informations sur le couplage interne entre la couche initiale et la partie reguli` ere de la solution.
Apr` es, la consistance des algorithmes mod` eles est consid´ er´ ee, suivie par une discussion de la stabilit´ e. En plus de d´ emontrer la stabilit´ e dans plusieurs cas (ce qui entraine aussi la convergence), le spectre de l’op´ erateur d’´ evolution discr` ete est examin´ e en d´ etail. Bas´ e sur ces r´ esultats il est montr´ e que la condition de CFL garantit la stabilit´ e dans le cas d’un algorithme ` a deux populations discr´ etisant l’´ equation d’advection. En outre, la discussion aborde la question de savoir jusqu’` a quel point il est possible d’obtenir des connaissances fiables sur le comportement de stabilit´ e par des arguments seulement fond´ es sur l’analyse formelle. Enfin, pour ´ elargir l’exp´ erience et pour pr´ eparer le travail futur, un exemple d’une couche de bord est analys´ e dans le contexte d’une discr´ etisation de diff´ erences finies pour l’´ equation de Poisson unidimensionnelle.
Zusammenfassung
Gitter-Boltzmann Methoden stellen eine verh¨ altnism¨ aßig neue Klasse numerischer Verfahren dar zur L¨ osung evolutionsartiger partieller Differentialgleichungen. Im Gegensatz zu Standardmethoden aus dem Bereich der finiten Differenzen bzw. finiten Elemente realisieren Gitter-Boltzmann Verfahren einen mesoskopischen (kinetischen) Ansatz. Die Kernidee besteht darin, eine gitterbasierten Pseudo-Teilchendynamik zu formulieren. Es stellt sich dabei heraus, daß gewisse gemittelte Gr¨ oßen die L¨ osungen bestimmter Differentialgleichungen approximieren, welche vor allem einem str¨ omungsmechanischen Hintergrund entstammen. Allerdings ist die Konsistenz der Gitter-Boltzmann Verfahren keineswegs offensichtlich, nicht zuletzt weil sie in enger Beziehung zu singul¨ ar skalierten Boltzmanngleichungen mit endlichen Geschwindigkeitsmodellen stehen.
Diese Arbeit besch¨ aftigt sich mit der Analyse von Gitter-Boltzmann Verfahren.
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Besonderes Augenmerk gilt dabei einigen “numerischen Ph¨ anomenen” wie dem Auftreten von Anfangsschichten, der Existenz mehrerer Zeitskalen und dem Zus-tandekommen von Randschichten. Beim Konsistenznachweis dienen regul¨ are asymptotische Entwicklungen (Hilbert Entwicklungen) als zentrales Hilfsmittel. Beispielhaft werden Gitter-Boltzmann Algorithmen in einer Raumdimension mit zwei und drei Populationen untersucht. Dabei wird zun¨ achst gezeigt, wie sich diese Modellal-gorithmen zur Diskretisierung der Advektions-Diffusions Gleichung aus zweidimensionalen Algorithmen unter Ausnutzung spezieller Symmetrieeigenschaften ergeben. Der Analyse der eigentlichen Schemata vorangestellt ist eine Untersuchung des singul¨ aren Grenzwerts bei einer Boltzmanngleichungen mit zwei bzw. drei Geschwindigkeiten. Alternativ lassen sich hier Konvergenzbeweise mittels einer Fourier-Entwicklung bzw. einer allgemeinen regul¨ aren Entwicklung kombiniert mit einer Energieabsch¨ atzung erzielen. Anfangsschichten werden mittels irregul¨ arer Entwicklungen bzw. Multiskalen-Entwicklungen aufgel¨ ost. Unter anderem st¨ oßt man dabei auf eine Hierarchie von Gleichungen, welche Aufschluß ¨ uber die interne Kopplung
der Anfangsschicht mit dem regul¨ aren Teil der L¨ osung geben. Anschließend wird die Konsistenz der Modellalgorithmen betrachtet, gefolgt von einer Stabilit¨ atsanalyse. Neben etlichen Stabilit¨ atsbeweisen (woraus Konvergenz der jeweiligen Verfahren gefolgert werden kann) wird das Spektrum des diskreten Evolutionsoperators einer genauen Untersuchung unterzogen. Darauf aufbauend l¨ aßt sich zeigen, daß sich die CFL-Bedingung sowie Stabilit¨ at im Falle eines Zwei-Populationen Algorithmus f¨ ur die Advektionsgleichung gegenseitig bedingen. Außerdem wird die M¨ oglichkeit er¨ ortert, inwieweit verl¨ aßliche Stabilit¨ atsaussagen auch anhand einer formalen Analyse gewonnen werden k¨ onnen. Um Erfahrung mit numerischen Randschichten f¨ ur zuk¨ unftige Untersuchungen zu sammeln, wird abschließend eine finite Differenzen Diskretisierung f¨ ur die eindimen- sionale Poisson Gleichung betrachtet, welche eine Randschicht erzeugt.
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Background and Outline
General framework and context
While computers were rapidly evolving since the sixties of the last century, scientists became aware of the opportunity to employ these powerful machines to simulate physical processes. Often, simulations are based on the numerical and hence approximate solution of certain differential equations, which form the core of a simplifyingmathematical model,abstracting the concrete process in a precise mathematical language.
In particular, the simulation of fluid flows (CFD1) represents a vast field of mathematically and numerically challenging problems whose mastering is of great practical interest for engineering and environmental sciences.
The underlying model equations used in CFD are taken fromcontinuum mechanicsandthermodynamics,see e.g. [18, 11]. By formulating the physical conservation principles for mass, momentum and energy in terms of familiar macroscopic quantities like pressure, flow velocity or temperature, the fundamental equations ofEulerandNavier-Stokesare found.
Since, in general, one cannot directly2solve differential equations arising in mathematical models, the equations must undergo somediscretization.This procedure breaks differential equations down into discrete equations being finally solvable by computers. Nowadays, various kinds of discretization methods are at disposal, among whichfinite differencesrepresent the historically oldest approach. Their intuitive idea consists in considering the solution (e.g. the velocity or pressure field) only in a finite number of time and space points. In contrast,finite elementandspectral methodstry to approximate the complicated solution in terms of comparatively simple functions, that are defined in the entire space domain but can be characterized by only a few numbers.
The fundamental equations, that are generally accepted to govern fluid motion, were deduced in the 18thand 19thcentury when only little was known about the internal structure of matter. Their derivation is based on thecontinuum hypothesis3, which
1Computational Fluid Dynamics
2To describe the solutions completely, infinitely (even uncountably) many numbers would be necessary which even a supercomputer cannot keep in its large but limited memory.
3The continuum hypothesis is a trick to deal with very large numbers of particles: for example, a large set of coupled oscillators is more easily described by the wave equation than by a huge
ODE system. Even in statistical mechanics the phase space density is introduced by this reason, which, however, might be alternatively interpreted as probability density.
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considers matter as a continuum, filling up a continuous space and moving in a continuously elapsing time. Even today, the continuum hypothesis has not lost its importance. Numerous differential equations of mathematical physics, whose validity has been well justified by experiments, rely on this basic assumption. Nevertheless, we now think of matter as composed of tiny, microscopic particles calledmoleculesandatoms.So nature seems ultimately to be discrete even if these particles exist in overwhelming large numbers. By the end of the 19thcentury this discovery led to the development ofstatistical mechanicsdealing with the be-havior of many particle systems. This new physical discipline unsealed another understanding of fluid motion from a quite different perspective. By means of the Boltzmann equation it was shown how conservation laws on the scale ofmicroscopicparticles give rise tomacroscopicconservation principles expressed by the Euler and Navier-Stokes equation.
In the early days of computer simulations, some scientists4started to study also purelydiscrete dynamical systems.Thesecellular automata(see [64, 65, 66] and [21]) were devised as some sort of counterpart to the discretization ofcontinuous dynamical systemsdescribed by differential equations. Although the first cellular automata seem like mathematical gadgets, there was a serious objective behind. The goal was to set up an artificial particle dynamics where the interaction of the particles (collisions) obeys simple rules while the system as a whole displays a rather complex behavior revealing also macroscopic regularities. In 1973 alattice-gas cellular automatonwas proposed by Hardy, Pomeau and de Pazzis [22, 23] that was able to produce flow like patterns in two space dimensions. However, it was shown that the results disagreed with the Navier-Stokes equation taken as reference whose accuracy for standard flow regimes is recognized. Only in 1986 Frisch, Hasslacher and Pomeau [20, 19] succeeded by considering a modified lattice-gas automaton which uses a hexagonal instead of a quadratic velocity model.
Now the development began to accelerate. In 1988 McNamara and Zanetti [47] replaced the boolean variables representing pseudo-particle numbers by real-valued quantities interpreted as particle distribution functions. During the following years, the basic lattice-Boltzmann algorithm received the form being still in use today [27], including the introduction of the BGK collision operator and the equilibrium function [51]. So, the essential steps were done which emancipated lattice-Boltzmann methods from their lattice-gas precursors, that were suffering from statistical noise because of the averaging to compute macroscopic quantities. In particular, relaxation type collision operators (BGK, MRT) do no more define explicit collision rules.
Formally, the classical lattice-Boltzmann method has much in common with a special finite difference scheme discretizing the BGK Boltzmann equation of statistical mechanics [24, 26]. Here the circle closes: departing from a general but blurred principle5and an artificial, rather unphysical particle dynamics, one had somehow
4Among them were mathematicians like von Neumann, Ulam and the computer pioneer Zuse.
5Amicroscopicallysimple dynamics can generate complex and well organized structures on amacroscopicscale.
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returned to the “real physics” described by the Boltzmann equation.
Much progress took place since the early nineties when the method came up. Today, lattice-Boltzmann methods compete with traditional methods of CFD and they are implemented in computational software tools for industrial purposes6with applications to a large range of flow problems7.
In comparison with other methods, the coding of lattice-Boltzmann schemes is less complex; moreover the code is well suited for parallelization to gain a speed-up of the computational time. These advantages are certainly owed to thebottom-upconception, which exploits the simplicity of the dynamics on the microscopic scale. In contrast, traditional methods are designed intop-downmanner departing from complicated macroscopic models which have to be ‘simplified’ then by discretization procedures.
In opposition to numeric methods like finite elements, being well established in mathematical research, lattice-Boltzmann methods are mainly developed by physicists and physical engineers. Since the theoretical foundations were consequently embedded in statistical mechanics, lattice-Boltzmann methods became especially appealing for researchers with physical background. This accounts for the origin of certain analytical techniques still widely adopted. Most notably we mention theChapman-Enskog expansionwith two time scales (see e.g. [25, 63]), which is employed as major tool to relate lattice-Boltzmann schemes with macroscopic equations like the Navier-Stokes equation. For the classical approach to lattice-Boltzmann methods we refer also to [14, 3, 9, 10, 30].
Although a fundamental understanding of lattice-Boltzmann methods has been reached utilizing these techniques, the explanations often are not satisfying and lack mathematical rigor. To overcome the mysticism occasionally attributed to lattice-Boltzmann methods, it is necessary to interpret them from different sides. So [46] advocates to give up a too ‘narrow viewpoint’ (which classifies lattice-Boltzmann methods just as derivatives of lattice-gas automata) in favor of a broader understanding. The paper emphasizes the affinity between lattice-Boltzmann methods and the kinetic theory of Boltzmann-type equations based on discrete velocity models. The latter has become a field of intensive research (see for instance [49]). Still there is not much literature available dealing with convergence, consistency and stability of lattice-Boltzmann methods. A first approach is found in [16] where the convergence of certain lattice-Boltzmann schemes with respect to a nonlinear diffusion equation and the viscous Burgers equation is shown. However, the considered collision operator admits anH-theoremwhich cannot be proven for collision operators of relaxation type and polynomial equilibria [68, 69] as used today. To avoid the relatively cumbersome Chapman-Enskog expansion, other possibilities have been introduced to connect a given lattice-Boltzmann algorithm with its macroscopic target equation (consistency analysis). In particular we mention
6For instance: theParPaccode at the Fraunhofer ITWM Kaiserslautern, or PowerFLOW commercialized by Exa corporation and based on “digital physics”.
7Thermal flow, non-Newtonian fluids, flow through porous media, shallow water, acoustics, multiphase and turbulent flows. For references consult [63].
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[33, 32], where the standard D2P9 lattice-Boltzmann algorithm was uncovered as a sophisticated finite difference discretization of the Navier-Stokes equation. Meanwhile, another approach has been presented in [34], which turns out to be especially clear and elegant, stimulating further developments of lattice-Boltzmann methods. Similar approaches are also found in [28, 15]. Furthermore it is tried8to interpret lattice-Boltzmann algorithms in the framework of finite volume schemes.
There are only few articles concerned with stability of lattice-Boltzmann schemes. Besides some publications examing stability and accuracy (e.g. [52]) by numerical tests, we mention [59] and [43] where a von Neumann stability analysis is essentially performed. Rigorous mathematical analysis is done in [1, 40].
Specific motivation of the thesis
The idea to this dissertation began to form, when I was working on grid coupling9and local grid refinement for lattice-Boltzmann methods. Being a novice in this field, I gradually became aware, that lattice-Boltzmann methods were mostly based on plausibility arguments being not proven in the mathematical sense. However, despite of being founded on plausible reasoning, some of my ideas did not work out in a fully satisfactory way. Because of these reasons the wish consolidated to acquire a deeper comprehension of lattice-Boltzmann methods. In particular, certain numerical phenomena like initial layers aroused my interest. Often, plausibility arguments tend to be superficial thereby risking to neglect crucial details. Hence numerical side effects, that might spoil the accuracy or even excite unstable modes of the scheme, may easily be overlooked or underestimated with respect to their impact.
The main objective of this thesis is to promote a better understanding of lattice-Boltzmann methods, where some attention is given to disturbing numerical effects. Frequently, theoretic investigations are faced with the difficulty that they should stick to problems of practical relevance on the one hand, while requiring a simplified setup to be analytically feasible on the other hand.
Here the focus is on theprincipalcomprehension which is to be obtained by adetailedstudy of several selected model problems, illuminating aspects also arising in more general situations. Thanks to their reduced complexity, the model problems permit a clear analysis and thereby offer much insight into the intrinsic workings of lattice-Boltzmann methods. Nevertheless, to bridge the gap mentioned above at least partly, it is shown how the one-dimensional model schemes may be derived as special cases from two-dimensional schemes.
Although lattice-Boltzmann methods are sometimes publicized as being fundamentally different from traditional numerical methods, they should be considered as just another discretization for numerically solving certain partial differential equations.
8ICMMES conference 2006, Hampton (VA). See also [62] for the one-dimensional case.
9Confer [54]. Observe, that the derivation of moments concerning the D2P9 algorithm for Stokes flow simulations as announced in this paper, is not included in this thesis to avoid further growing of its extent.
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Figure 1:Phenomenological description of an initial layer.The curves in the left diagram represent the numerical error of a Stokes flow simulation performed with the D2P9 lattice-Boltzmann algorithm. More precisely, the curves indicate the relativeL1-error in thex-componentof the flow velocity plotted versus the time. The simulations were executed on three different grids. Two observations are striking: First, the error seems to be quartered if the grid spacinghis halved, which suggests that the error is of magnitudeO(h2).Second, the error oscillates at the beginning, where the amplitude (attenuation) is the smaller (stronger), the finer the grid is chosen. These features are typical for an initial layer combining two time scales here. The discrete time scale is manifested by the damping and the oscillations from time step to time step, being hardly visible due to the low resolution of the figure. In contrast, the beat-bellies occur in the fast time scale, which is slower than the discrete time scale but faster than the time scale referring to the labels of the horizontal axis. Considering the time evolution of an arbitrary single population in a fixed node reveals similar oscillations. This indicates that the initial layer affects all populations in roughly the same manner and does not represent an integral phenomenon only appearing in theL1-norm. The depicted initial layer was observed by simulating a Taylor vortex whose streamlines are given at right. Taylor vortices actually correspond to the eigenmodes of the Stokes operator in a periodic, rectangular domain. Therefore the streamlines remain invariant in space while the stream function decreases exponentially in time. Hence the error should drop likewise. Here, however, this decrease is imperceptible over the simulated time interval because of the relatively small viscosity that was chosen to sustain the initial layer excited by a crude initialization.
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However, two particular features must be taken into account:
•The most striking distinction is that the physical quantities of the macroscopic limit equations do not figure as primary variables (populations) in the lattice-Boltzmann algorithms. Besides, there are more primary variables than relevant physical quantities.
•Viewing lattice-Boltzmann algorithms in the light of standard finite difference schemes, they look much like the natural discretization of Boltzmann-type equations so that generally no direct relation with the actual limit equation is suggested.
Roughly speaking, a numerical scheme is calledconsistentwith respect to a differential equation if the solution of the differential equation satisfies the discrete equations in every grid node up to a small residual. Due to the first item, the standard notion of consistency does not apply to lattice-Boltzmann schemes. However, extending this notion, one is enabled to examine the consistency of a given lattice-Boltzmann method by means ofregular expansions.
The resulting analysis shows why the primary variables yield approximate solutions of the limit equation. Additionally, it is found that they encode further information concerning the differentiated solution of the limit equation. This circumstance may justify the enhanced memory requirements which lattice-Boltzmann methods need in comparison with other methods.
If compared with the twoscale Chapman-Enskog expansion, regular asymptotic expansions (truncated power series) seem to be a much more natural approach to analyze lattice-Boltzmann methods. In fact, a regular expansion was already used by Hilbert10to study the scaled Boltzmann equation. Furthermore, in numerics, regular expansions date back to early investigations of finite difference methods, to specify the numerical error (see e.g. [50] page 135). Nonetheless, only recently this technique has been applied to lattice-Boltzmann methods [34], which is also adopted here.
Let us finally remark with regard to the second item that under certain scalings, solutions of Boltzmann-type equations display a convergence behavior with respect to other differential equations. This indicates that the convergence of lattice-Boltzmann schemes encapsulates two limit processes:
•The convergence of the scheme under the refinement of the discretization.
•The singular limit inherent in scaled Boltzmann-type equations.
Therefore typical phenomena of scaled Boltzmann-type differential equations can be observed likewise in the context of lattice-Boltzmann algorithms. Among other topics we will particularly focus on this issue which includes the appearance of multiple time scales and initial layers (see figure 1).
10That is why it is often referred to as Hilbert expansion in the context of kinetic theory.
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Organization and scope of the work
Prerequisites concerning some specific vocabulary, notation and the general strategy of convergence proofs are collected in subsection 1.1.2 (concerning lattice-Boltzmann methods) and section 1.3 (concerning the numeric/asymptotic analysis). Apart from this, the chapters as well as the three sections of the first chapter are nearly self-contained and should be readable independently from each other. The cohesion is mainly generated by the circumstance that different aspects of equal or similar objects are discussed over several chapters.
To keep the notation simple, there is no unique notation for the entire thesis. Instead, the notation may slightly change from chapter to chapter or even from section to section.
Passages appearing in small printing are less important. This concerns digressions providing extra-material, some lemmas and propositions having rather the character of a tool than a result in the specific context as well as some tedious computations. Furthermore the text is complemented by numerous footnotes containing additional explanations, suggestions and cross-references.
To facilitate the orientation, a summary of each chapter is given here:
Chapter 1conveys some background knowledge about the Boltzmann equation insubsection 1.1.1.Above all, the provenance of the Boltzmann equation is motivated and its relation to macroscopic equations of fluid dynamics is established. In contrast to the historic development shortly outlined above,subsection 1.1.2introduces to lattice-Boltzmann methods from the perspective of mathematical kinetic theory dealing with Boltzmann-type equations that are much simpler than the original Boltzmann equation but retain essential properties. It is shown that scaled finite-velocity Boltzmann equations (also referred to as lattice-Boltzmann equations) reveal much flexibility to be related with other equations via limit processes. Finally, it is shown how the classical lattice-Boltzmann scheme is derived from finite-velocity Boltzmann equations.
A rather technical paragraph is concerned with the computation of the structure relations pertaining to discrete velocity models. In contrast to one-dimensional velocity models, these relations get more complicated in two and more dimensions; concretely the D2P9 model is considered. Theses relations are essential prerequisites for the analysis of lattice-Boltzmann methods.
The goal ofsection 1.2is to underline the close relationship between lattice-Boltzmann schemes in two space dimensions (being already of practical significance) and the model algorithms to be investigated here.
So, the D1P3 lattice-Boltzmann algorithm is derived from the standard D2P9 al-gorithm. Combining symmetry properties of the D2P9 algorithm with special initializations, the number of populations as well as the number of spatial dimensions can be reduced such that the D1P3 algorithm is finally extracted. Endowed with a slightly more general equilibrium than found in this derivation, the D1P3 lattice-Boltzmann algorithm serves as our main model scheme.
Notice that the D1P2 algorithm considered in several chapters is a particular case of
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the D1P3 scheme. However, it could be derived independently of the D1P3 scheme by considering an appropriate D2P4 lattice-Boltzmann algorithm.
Insection 1.3we provide some concepts of numerical and asymptotical analysis in as much as they are needed here. Particular attention is paid to the notions ofconsistencyandstability.The abstract presentation is illustrated by examples.
Before we address lattice-Boltzmann algorithms in chapter 4, 5 and 6, we examine some properties of lattice-Boltzmann equations. Formally, the latter can be investigated more easily since asymptotic expansions with respect to the scaling parameter are applied without being combined with Taylor expansions. Many features of lattice-Boltzmann algorithms are not of specific discrete nature as shown in chapter 2 and 3. Examples are the hierarchy of asymptotic orders, the decoupling of odd and even orders, the occurrence of initial layer etc..
Chapter 2deals with a D1P2 lattice-Boltzmann equation. Insection 2.1the difference between thehyperbolicandparabolicscaling is highlighted. Furthermore, we show that the D1P2 lattice-Boltzmann equation, which is actually a system, can be transformed into a scalar, singularly perturbed equation. For the case, of the parabolic scaling and a linear equilibrium, convergence is demonstrated insection 2.2by expanding the solution in terms of a Fourier series. Generally, uniform convergence in time does not extend to the temporal derivative. This indicates the existence of an initial layer, which is expected whenever the solution of the perturbed equation is not initialized compatibly with the limit problem. Confining ourselves on a single Fourier mode insection 2.3,the appearance of the initial layer and its structure is thoroughly analyzed. A twoscale expansion discloses a hierarchy of equations for the initial layer and the regular part of the solution which are mutually coupled.
Chapter 3presents the analysis of a lattice-Boltzmann equation based on the D1P3 velocity model, which is distinguished from the D1P2 model by an additional rest population. In this case convergence is shown without recourse to an equivalent scalar reformulation. Instead, the proof is divided into a stability part accomplished by an energy estimate and a consistency part performed by a regular asymptotic expansion. It is observed that the regular expansion can only cover very special initial values. Therefore further, irregular expansions are introduced such that arbitrary initial values can be handled. These expansions turn out to describe initial layers. The chapter closes with a short discussion of non-periodic boundary conditions.
Chapter 4is devoted to the consistency analysis of a D1P3 lattice-Boltzmann algorithm discretizing the advection-diffusion equation. The algorithm is essentially the discrete counterpart of the lattice-Boltzmann equation studied in chapter 3. The regular expansion is formally performed up to arbitrary orders insection 4.1providing structural insight. Above all, it is seen how a hierarchy of evolution equations builds up, which determines the asymptotic order functions recursively. Furthermore the asymptotic expansion indicates the convergence order. Insection
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4.2,the consistency of truncated expansions is defined, which are then constructed insection 4.3.At last, we throw a quick glance at numerical initial layers displaying more structure in the discrete case.
Whereas the previous investigations were mainly concerned with lattice-Boltzmann equations and algorithms in parabolic scaling, a hyperbolically scaled D1P2 algorithm is discussed inchapter 5.The consistency analysis, which proceeds analogously to chapter 3 and 4, shows that the algorithm discretizes the advection equation. However, the major interest of this chapter is focused on the long-term behavior of the scheme which is wrongly predicted by the regular expansion. Actually the evolution of the algorithm contains different times scales which are detected by numerical experiments at the beginning ofsection 5.2.Using a multiscale expansion, the analysis of the algorithm yields a precise description of its behavior, which is validated by examples.
To justify the expansions in chapter 4 and 5 rigorously and to answer thereby the question of convergence, we examine thestabilityof the algorithms inchapter 6.After some preparatory remarks about shift matrices to write the algorithms in matrix form,ℓ∞-stability in time and space is proven insection 6.2.The result refers to the parabolically scaled D1P2 algorithm specializing the D1P3 algorithm in chapter 4. A detailed spectral analysis of the evolution matrix continues the section. Among others, eigenvectors and eigenvalues are computed where the latter are compared with the eigenvalues that belong to the evolution operator of the diffusion-advection equation. Furthermore the symmetry properties of the spectrum are discussed.
Section 6.3considers almost the same D1P2 algorithm as presented in chapter 5 but inhyperbolicscaling . Since this algorithm discretizes the advection equation, analyzing the role of the CFL-condition is natural. Using the results in subsection 6.2.2, it is shown that the CFL-condition is necessary and sufficient for stability. However, this statement cannot be extended to the analogous D1P3 scheme as pointed out in 6.5. The section is finished by comparing the actual stability properties with a formal stability discussion based on the multiscale expansion in 5.2.
So far, the considered algorithms refer to periodic boundary conditions. Insection 6.4we deal with bounce-back type boundary conditions emulating homogeneous Dirichlet and Robin boundary conditions of the macroscopic limit equation. For the diffusive equilibrium, eigenvalues and eigenvectors of the evolution matrix pertaining to the parabolically scaled D1P2 algorithm are computed. However, the consistency analysis of the boundary conditions is left for future work. Finally, insection 6.5we discuss the possibilities to generalize the results of 6.2 and 6.3 to the corresponding D1P3 algorithms. Furthermore we briefly resume the discussion of 6.4 where now an additional advective term is admitted in the equilibrium.
Since kinetic boundary conditions in two and more space dimensions generally produce boundary layers, we exemplarily study this phenomenon inchapter 7.Con- cretely, we consider the one-dimensional Poisson equation with Dirichlet boundary
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conditions. This is discretized by means of a five-point stencil where different extrapolations are used to deal with the overhanging parts of the stencil near the boundary. After some preliminaries, we prove stability of the resulting schemes insection 7.4.It turns out that stability alone is not enough to prove the observed convergence rates. For this, finer estimates are needed resulting from thedamping propertyofsection 7.5.The chapter closes with a presentation of numerical experiments that illustrate the irregular asymptotic expansion which describes the numerical error.
In theappendixwe compute the eigenvalues of the linear, one-dimensional advection-diffusion operator with constant coefficients for various boundary conditions. From this the eigenvalues of the semigroup of evolution operators associated to the advection-diffusion equation are easily obtained. They are compared with the eigenvalues of lattice-Boltzmann evolution matrices in chapter 6.
Chapter 5 was originally designed as manuscript for a shortcourse which took place at ICMMES112006 in Hampton (VA). It is also connected with [35] while section 6.3 is based on [55].
11International Conference of Mesoscopic Methods for Engineering and Science
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Chapter 1
Introduction to lattice Boltzmann
methods and their analysis
1.1 Initiation to lattice-Boltzmann methods
Lattice-Boltzmann methods can be approached from three major directions as depicted in figure 1.1:
•cellular automata, particularly lattice-gas automata,
•relaxation schemes and discretization methods (finite differences),
•Boltzmann equation and kinetic theory of gases.
Historically, lattice-Boltzmann methods attracted attention as an important improvement of lattice-gas automata. So they were not originally based on differential equations but rooted in the realm of discrete dynamical systems with boolean variables. From a mathematical point of view, lattice-Boltzmann equations are akin to relaxation schemes which becomes apparent if they are transformed into their equivalent moment system. Finally, a lot of inspiration and vocabulary - including the name lattice-Boltzmann - is borrowed from the theory twining around the Boltzmann equation. Unlike some textbooks introducing to lattice-Boltzmann methods (cf. [63], [60]), we shall not retrace the historic development and skip the field of lattice-gas automata completely.
This first section of chapter 1 begins with the physical background touching on some key-words in the field of kinetic theory (for instance: equilibrium, collision invariant, BGK, scaling). Subsection 1.1.1 leans partly on what may be found in textbooks about statistical physics and fluid dynamics; it is mostly drawn from equivalent sections of the books [7, 8]. The last paragraph gives a short survey of the scaled Boltzmann equation referring to the paper [2].
Subsection 1.1.2 imparts the essential ideas leading to “stylized forms of the Boltzmann equation now going by the namelattice-Boltzmann equation.These relinquish most mathematical complexities of the true Boltzmann equation without sacrificing physical fidelity in the description of many situations involving complex fluid motion1.”
1Text in quotation marks cited from the synopsis of [60].
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Figure 1.1:Lattice-Boltzmann methods embedded in their mathematical and physical context.
Furthermore, subsection 1.1.2 portrays the kinetic approach and its link to relaxation methods by a simple example. A technical paragraph collects some useful facts that are necessary for the analysis of 2D lattice-Boltzmann methods. Finally, the lattice-Boltzmann algorithm is derived from the lattice-Boltzmann equation.
There might be two principal reasons to deal with the Boltzmann equation and kinetic theory:
•First, the Boltzmann equation describes relevant physics.
•Second, the Boltzmann equation exhibits a rich structure which can be exploited to use it as an approximating tool for solving other equations.
As our objective is to pave a way towards lattice-Boltzmann methods, we take up the second perspective. Therefore we focus on structural aspects and the relation between the Boltzmann equation and macroscopic fluid dynamics. Above all, we do not discuss the intrinsics of the collision operator but recall only certain structurally important properties.
1.1.1 A brief introduction to kinetic theory
Kinetic theory is a branch of statistical physics, which is concerned withnonequilibriumphenomena especially in the context of fluids (e.g. gases and liquids). This definition obliges us to explain the term equilibrium more precisely: if a system is left without exterior interaction, it reaches an equilibrium state2after a certain while (relaxation time). In this final state the system can be characterized by only
2Equilibria are stationary states but the converse is not true. For instance, a rod between two heat sources of different temperature evolves into a stationary state with a constant temperature
gradient. If the rod is isolated from the heat sources the temperature relaxes gradually towards an equilibrium marked by a spatially constant temperature.
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1.1. Initiation to lattice-Boltzmann methods13
a few macroscopic quantities like pressure, temperature, concentration etc.. However, in most applications the system is exposed to outer influences resulting into more complicated non-equilibrium states, which also need an adequate description to become computable. For a concrete illustration we imagine a monoatomic gas enclosed inside a boxB ⊂R3.
Microscopic picture
The first approach describes the system on a microscopic level keeping track of every single particle. Each of theNatoms, belonging all to the same species, is modeled as a smallhard sphereof massmwithout any internal degrees of freedom. Hence possible self-rotation3, vibrational modes or the energetic excitation of the electronic shell are completely disregarded. Similarly to the balls of a billiard game, the atoms interact with each other byelastic collisionsconserving linear momentum and kinetic energy. Likewise, the atoms undergospecular4reflectionif hitting the wall. Mathematically, the gas can be described in the framework of classical mechanics by a system ofHamilton equations5reducing to the simple case of free motion
˙ ˙xivi,vii∈ {1,..., N},
complemented by additional collision6and reflection7rules. Herexidenotes the position andvithe velocity of theithparticle as a function of time. Between collisions, the velocitiesviremain constant and the particles move uniformly. The state of the whole system corresponds to a point in a 6N -dimensional state
3In this model a hard sphere particle has no orientation like a rigid body. Even if it could rotate about some axis, there is no energy exchange admitted by collisions between the rotational and
translational degrees of freedom. This would require tangential contact forces (friction). As any interchange between translational and rotational energy is excluded, the rotational energy matters
only as a constant, which is conveniently set to zero (free choice of energy scale).
4Derived from the Latin wordspeculum
5For autonomous systems the Hamilton function represents the total energy in terms of spatial coordinatesxiand generalized momentapi.The equations of motion (Hamilton equations) are
˙ ˙xi∇piH,pi−∇xiH, i∈ {1,..., N}.PN
i=1p2The Hamilton function for the system ofNhard spheres isH(x1, ...,xN,p1,...,pN1i,
2m
wherepimviandvidenotes the velocity of theithsphere.
6If two hard spheres of equal mass (but not necessarily of equal volume) collide with each other, their post-collisional velocities are given in terms of their pre-collisional velocities by
vpostvpre+vpre−vpre 1,ee,
1 1 2vpostvpre− vpre−vpre 1,ee.
2 2 2
The unit vectorepoints from the center of the first to the center of the second sphere at the
moment of collision. Evidently the exchange of momentum (velocity) occurs only in the direction
ofe.
7Ifndenotes the unit normal vector of the reflecting surface, the outgoing velocityvoutof the reflected particle is given in terms of the incoming velocityvinby the formula
voutvin+ 2vin,nn.
Herenis assumed to be directed into the half-space of the particle, i.e.vin,n≤0. This formula describes (specular) reflection.
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1.1. Initiation to lattice-Boltzmann methods15
far beyond feasibility to handle the enormous amount of arising data. Moreover, the exact movement of each particle is of little interest, because it does not significantly affect the state of the entire system. Therefore a so-calledmesoscopicapproach has been devised which does not consider individual particles as in the abovemicroscopicdescription.
Motivation of the Boltzmann equation
To obtain the mesoscopic description, the high-dimensional Γ-space is reduced to the six-dimensional phase space (µ-space10) of a single particle. Analogously, the functionPwith 6N + 1 arguments becomes a function (t,x, v)→ f(t,x, v)of only seven arguments, wheretstands for the time andx, vcontain each three components to fix the position and velocity. Similarly toPthe distribution functionfhas a probabilistic meaning depending on how it is gauged:
ForX⊂ BandV⊂R3the integral
f(t,x, v)dv dxX V
permits the following interpretations:
i) probability that allNparticles are inX×V,
ii) expected number of particles inX×V,
iii) expected mass inX×V.
Here we prefer the third interpretation. Iff(t,x, v)is integrated over the entire velocity space it gives the local mass density in space.
The evolution of the distribution functionfis prescribed by theBoltzmann equation
∂tf+v· ∇xfQ(f, f) (1.1)
adopting the role of Liouville’s equation. The Boltzmann equation becomes quickly comprehensible by a short reasoning of balance: Consider for this a small volumeX×Varound a point (x,v)in the phase-space. Assume thatfis scaled so that the integral over the whole phase space corresponds to the number of particles. By what effects will the number of particles change inX×Vover a short time? First, some particles leaveXwhile other enter it from the surrounding. This is taken into account by the advective termv· ∇xf. There is a total loss of particles (with velocities inVclose tov)ifvis directed parallel to the gradient. Then more particles move intoXthan stream out. Otherwise, in the case of antiparallelvand∇xf, the reverse process happens andXexperiences a total gain of particles.
10A mnemonic might be:µabbreviating the Greek wordµ´ oνoς
