# Analysis of Lattice-Boltzmann Methods

## Asymptotic and numeric investigation of a singularly perturbed system

### 377 Pages

Excerpt

Analysis of Lattice-Boltzmann Methods

Asymptotic and Numeric Investigation of a
Singularly Perturbed System

Martin Kilian Rheinläander

Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr. rer. nat.)
vom Fachbereich Mathematik und Statistik
der Universität Konstanz genehmigte Dissertation.
Datum der Präsentation: 08.05.07
Datum des Rigorosums: 25.07.07

Preface – Vorwort

[...]

Zusammenfassung

Gitter-Boltzmann Methoden stellen eine verhältnismäßig neue Klasse numerischer Verfahren dar zur Lösung 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 Größen die Lösungen bestimmter Differentialgleichungen approximieren, welche vor allem einem strömungsmechanischen Hintergrund entstammen. Allerdings ist die Konsistenz der Gitter-Boltzmann Verfahren keineswegs offensichtlich, nicht zuletzt weil sie in enger Beziehung zu singulär skalierten Boltzmanngleichungen mit endlichen Geschwindigkeitsmodellen stehen.

Diese Arbeit beschäftigt sich mit der Analyse von Gitter-Boltzmann Verfahren.

Besonderes Augenmerk gilt dabei einigen “numerischen Phänomenen” wie dem Auftreten von Anfangsschichten, der Existenz mehrerer Zeitskalen und dem Zustandekommen von Randschichten. Beim Konsistenznachweis dienen reguläre asymptotische Entwicklungen (Hilbert Entwicklungen) als zentrales Hilfsmittel. Beispielhaft werden Gitter-Boltzmann Algorithmen in einer Raumdimension mit zwei und drei Populationen untersucht. Dabei wird zunächst gezeigt, wie sich diese Modellalgorithmen zur Diskretisierung der Advektions-Diffusions Gleichung aus zweidimensionalen Algorithmen unter Ausnutzung spezieller Symmetrieeigenschaften ergeben. Der Analyse der eigentlichen Schemata vorangestellt ist eine Untersuchung des singulären Grenzwerts bei einer Boltzmanngleichungen mit zwei bzw. drei Geschwindigkeiten. Alternativ lassen sich hier Konvergenzbeweise mittels einer Fourier- Entwicklung bzw. einer allgemeinen regulären Entwicklung kombiniert mit einer Energieabschätzung erzielen. Anfangsschichten werden mittels irregulärer Entwicklungen bzw. Multiskalen-Entwicklungen aufgelöst. Unter anderem stößt man dabei auf eine Hierarchie von Gleichungen, welche Aufschluß über die interne Kopplung der Anfangsschicht mit dem regulären Teil der Lösung geben.

Anschließend wird die Konsistenz der Modellalgorithmen betrachtet, gefolgt von einer Stabilitätsanalyse. Neben etlichen Stabilitätsbeweisen (woraus Konvergenz der jeweiligen Verfahren gefolgert werden kann) wird das Spektrum des diskreten Evolutionsoperators einer genauen Untersuchung unterzogen. Darauf aufbauend läßt sich zeigen, daß sich die CFL-Bedingung sowie Stabilität im Falle eines Zwei- Populationen Algorithmus für die Advektionsgleichung gegenseitig bedingen. Außerdem wird die Möglichkeit erörtert, inwieweit verläßliche Stabilitätsaussagen auch anhand einer formalen Analyse gewonnen werden können.

Um Erfahrung mit numerischen Randschichten für zukünftige Untersuchungen zu sammeln, wird abschließend eine finite Differenzen Diskretisierung f¨ur die eindimensionale Poisson Gleichung betrachtet, welche eine Randschicht erzeugt.

Contents

Background and Outline ... 1

1 Introduction to lattice Boltzmann methods and their analysis ...  11
1.1 Initiation to lattice-Boltzmann methods ...  11
1.1.1 A brief introduction to kinetic theory ... 12
1.1.2 A primer of lattice-Boltzmann methods ...  26
1.2 Translational invariance and dimensional reduction  ...  43
1.3 An abstract framework of numerical analysis  ... 55

2 Scalings and singular limits on the basis of the D1P2 model  ... 71
2.1 Hyperbolic versus parabolic scaling  ...  72
2.2 A singularly perturbed initial value problem ... 81
2.2.1 The Fourier coefficient functions ... 82
2.2.2 Solution of the perturbed problem and convergence  ...  92
2.2.3 Uniform convergence and convergence rate ...  96
2.3 Two-scale expansion and resolution of the initial layer  ... 108

3 Analysis of a D1P3 lattice-Boltzmann equation  ... 127
3.1 Energy estimate and stability ... 129
3.2 Regular expansion and consistency  ... 133
3.3 Smoothness conditions and convergence  ... 140
3.4 Initial conditions and irregular expansions  ...  145
3.5 A glimpse of boundary conditions  ... 149

4 Consistency of a D1P3 lattice-Boltzmann algorithm  ... 157
4.1 Formal expansion  ... 160
4.2 Consistency and asymptotic similarity  ... 172
4.3 Construction of consistent population functions  ...  178
4.4 Initial behavior  ...  188

5 Long-term behavior of an advective lattice-Boltzmann scheme  ... 197
5.1 Regular expansion  ...  200
5.1.1 Analysis of the update rule  ...  200
5.1.2 Smooth initialization and consistency ...  207
5.2 Multiscale Expansion ...  213
5.2.1 A numeric test to detect different time scales  ... 213
5.2.3 Emergence of a cubic time scale ...  220

6 Stability investigations around the D1P2 model  ... 227
6.1 Basics concerning shift matrices  ... 228
6.2 LB advection-diffusion scheme with periodic boundary conditions ...  231
6.2.1 An .(?)‡-stability result  ...  235
6.2.2 The spectral limit set of the evolution matrices  ...  239
6.2.3 Asymptotics and symmetry of eigenvalues  ...  250
6.3 LB advection scheme with periodic boundary conditions  ...  258
6.3.1 The CFL-condition and stability  ...  260
6.3.2 Stability in the .2-norm  ... 264
6.3.3 Multiscale expansion and stability ... 268
6.4 LB diffusion scheme with bounce-back type boundary conditions  ...  272
6.4.1 Evolution matrices and their spectra  ...  274
6.4.2 Computing eigenbases ...  283
6.5 Towards the D1P3 scheme & Concluding remarks  ...  290

7 Asymptotic analysis of a numeric boundary layer  ... 299
7.1 Some remarks about interpolation and difference stencils  ... 301
7.2 Model problem: 1D Poisson equation with Dirichlet BC ... 308
7.3 Discretization of Dirichlet boundary conditions ... 311
7.4 Stability of extrapolation schemes ... . 316
7.5 Damping property of discrete inverse operators ... 325
7.6 Asymptotic expansions and convergence  ... 330
7.7 Numeric experiments ...  340

A Appendix  ... 351
A.1 Eigenvalues of the advection-diffusion operator ...  351
Bibliography  ... 361

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 simplifying mathematical 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 from continuum mechanics and thermodynamics, 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 of Euler and Navier-Stokes are found.

Since, in general, one cannot directly2 solve differential equations arising in mathematical models, the equations must undergo some discretization. This procedure breaks differential equations down into discrete equations being finally solvable by computers. Nowadays, various kinds of discretization methods are at disposal, among which finite differences represent 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 element and spectral methods try 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 18th and 19th century when only little was known about the internal structure of matter. Their derivation is based on the continuum hypothesis3, which 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 called molecules and atoms. So nature seems ultimately to be discrete even if these particles exist in overwhelming large numbers. By the end of the 19th century this discovery led to the development of statistical mechanics dealing with the behavior 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 of microscopic particles give rise to macroscopic conservation principles expressed by the Euler and Navier-Stokes equation.

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 called molecules and atoms. So nature seems ultimately to be discrete even if these particles exist in overwhelming large numbers. By the end of the 19th century this discovery led to the development of statistical mechanics dealing with the behavior 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 of microscopic particles give rise to macroscopic conservation principles expressed by the Euler and Navier-Stokes equation.

In the early days of computer simulations, some scientists4 started to study also purely discrete dynamical systems. These cellular automata (see [64, 65, 66] and ) were devised as some sort of counterpart to the discretization of continuous dynamical systems described 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 a lattice-gas cellular automaton was 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  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 , including the introduction of the BGK collision operator and the equilibrium function . 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.

[....]

1 Computational Fluid Dynamics

2 To describe the solutions completely, infinitely (even uncountably) many numbers would be necessary which even a supercomputer cannot keep in its large but limited memory.

3 The 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 dens

4 Among them were mathematicians like von Neumann, Ulam and the computer pioneer Zuse.

Excerpt out of 377 pages

Details

Title
Analysis of Lattice-Boltzmann Methods
Subtitle
Asymptotic and numeric investigation of a singularly perturbed system
College
University of Constance  (Fachbereich Mathematik & Statistik)
Author
Year
2007
Pages
377
Catalog Number
V79991
ISBN (eBook)
9783638784757
ISBN (Book)
9783638796064
File size
4926 KB
Language
English
Notes
Die Arbeit wurde mit 1 (magna cum laude bewertet) und enthält farbige Abbildungen.
Tags
Analysis, Lattice-Boltzmann, Methods
Quote paper
Martin Rheinländer (Author), 2007, Analysis of Lattice-Boltzmann Methods, Munich, GRIN Verlag, https://www.grin.com/document/79991 