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.
Contents
1 Introduction to lattice Boltzmann methods and their analysis
1.1 Initiation to lattice-Boltzmann methods
1.1.1 A brief introduction to kinetic theory
1.1.2 A primer of lattice-Boltzmann methods
1.2 Translational invariance and dimensional reduction
1.3 An abstract framework of numerical analysis
2 Scalings and singular limits on the basis of the D1P2 model
2.1 Hyperbolic versus parabolic scaling
2.2 A singularly perturbed initial value problem
2.2.1 The Fourier coefficient functions
2.2.2 Solution of the perturbed problem and convergence
2.2.3 Uniform convergence and convergence rate
2.3 Two-scale expansion and resolution of the initial layer
3 Analysis of a D1P3 lattice-Boltzmann equation
3.1 Energy estimate and stability
3.2 Regular expansion and consistency
3.3 Smoothness conditions and convergence
3.4 Initial conditions and irregular expansions
3.5 A glimpse of boundary conditions
4 Consistency of a D1P3 lattice-Boltzmann algorithm
4.1 Formal expansion
4.2 Consistency and asymptotic similarity
4.3 Construction of consistent population functions
4.4 Initial behavior
5 Long-term behavior of an advective lattice-Boltzmann scheme
5.1 Regular expansion
5.1.1 Analysis of the update rule
5.1.2 Smooth initialization and consistency
5.2 Multiscale Expansion
5.2.1 A numeric test to detect different time scales
5.2.2 Additional quadratic time scale
5.2.3 Emergence of a cubic time scale
6 Stability investigations around the D1P2 model
6.1 Basics concerning shift matrices
6.2 LB advection-diffusion scheme with periodic boundary conditions
6.2.1 An ℓ∞-stability result
6.2.2 The spectral limit set of the evolution matrices
6.2.3 Asymptotics and symmetry of eigenvalues
6.3 LB advection scheme with periodic boundary conditions
6.3.1 The CFL-condition and stability
6.3.2 Stability in the ℓ2-norm
6.3.3 Multiscale expansion and stability
6.4 LB diffusion scheme with bounce-back type boundary conditions
6.4.1 Evolution matrices and their spectra
6.4.2 Computing eigenbases
6.5 Towards the D1P3 scheme & Concluding remarks
7 Asymptotic analysis of a numeric boundary layer
7.1 Some remarks about interpolation and difference stencils
7.2 Model problem: 1D Poisson equation with Dirichlet BC
7.3 Discretization of Dirichlet boundary conditions
7.4 Stability of extrapolation schemes
7.5 Damping property of discrete inverse operators
7.6 Asymptotic expansions and convergence
7.7 Numeric experiments
Research Objectives and Topics
This thesis aims to provide a rigorous mathematical understanding of Lattice-Boltzmann methods, which are numerical schemes for solving evolutionary partial differential equations (PDEs). The central research question explores how Lattice-Boltzmann algorithms approximate PDEs and how numerical phenomena such as initial layers, boundary layers, and multiple time scales affect their stability and consistency.
- Rigorous analysis of Lattice-Boltzmann methods using regular and irregular asymptotic expansions (Hilbert expansion).
- Investigation of singular limits in scaled finite-velocity Boltzmann equations.
- Stability analysis of various algorithms, including spectral analysis and the role of the CFL-condition.
- Examination of numerical boundary layers, particularly for the one-dimensional Poisson equation.
- Bridging the gap between mesoscopic kinetic theory and macroscopic fluid dynamics.
Auszug aus dem Buch
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 relative L1-error in the x-component of 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 spacing h is halved, which suggests that the error is of magnitude O(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 the L1-norm.
Summary of Chapters
Chapter 1: Provides fundamental background on kinetic theory, the Boltzmann equation, and an introduction to the Lattice-Boltzmann method's mathematical framework.
Chapter 2: Analyzes the D1P2 Lattice-Boltzmann model, focusing on the distinction between hyperbolic and parabolic scaling and the investigation of initial layers.
Chapter 3: Presents an analysis of a D1P3 Lattice-Boltzmann equation, utilizing energy estimates and asymptotic expansions to establish convergence and stability.
Chapter 4: Conducts a formal consistency analysis of a D1P3 Lattice-Boltzmann algorithm discretizing the advection-diffusion equation, highlighting the recursive hierarchy of evolution equations.
Chapter 5: Discusses the long-term behavior of an advective Lattice-Boltzmann scheme, identifying different time scales using multiscale expansions.
Chapter 6: Investigates the stability of D1P2 models through spectral analysis of evolution matrices and examines the role of boundary conditions.
Chapter 7: Performs an asymptotic analysis of numerical boundary layers using the Poisson equation as a model problem, emphasizing the damping properties of discrete inverse operators.
Keywords
Lattice-Boltzmann methods, kinetic theory, asymptotic expansion, numerical stability, partial differential equations, initial layers, boundary layers, convergence analysis, singular limits, Hilbert expansion, Chapman-Enskog expansion, finite difference schemes, spectral analysis, grid refinement, computational fluid dynamics.
Frequently Asked Questions
What is the fundamental focus of this research?
The work focuses on the mathematical analysis of Lattice-Boltzmann algorithms to understand their consistency, convergence, and stability when solving evolutionary partial differential equations.
Which scientific methods are primarily employed?
The research relies heavily on asymptotic analysis, specifically using regular and irregular (Hilbert) expansions, multiscale expansions, and spectral analysis of discrete evolution operators.
What are the central thematic areas covered in this document?
The document covers kinetic theory, the derivation of Lattice-Boltzmann methods from Boltzmann equations, singular limits, the impact of initial and boundary layers, and the stability of these numerical schemes.
What is the primary research goal?
The main objective is to provide a rigorous mathematical foundation for Lattice-Boltzmann methods, moving beyond purely plausibility-based arguments to improve the reliability of these simulations in practical applications.
How is the consistency of the schemes analyzed?
Consistency is established by utilizing regular asymptotic expansions to derive how the discrete Lattice-Boltzmann equations approximate the macroscopic target equations, such as the Navier-Stokes or advection-diffusion equations.
Which specific "numerical phenomena" are identified and analyzed?
The thesis investigates numerical phenomena such as initial layers, boundary layers, and the presence of multiple time scales that can interfere with the accuracy and stability of the algorithms.
How do different velocity models (D1P2, D1P3, D2P9) relate to each other?
The thesis demonstrates that higher-dimensional or more complex models, such as the D2P9, can be reduced to simpler one-dimensional models like D1P3 or D1P2 under certain symmetry and initialization conditions.
What is the significance of the "damping property" mentioned in the later chapters?
The damping property is a crucial condition required for analyzing the stability of discrete inverse operators, particularly when dealing with singular residuals in boundary layer analysis.
- Quote paper
- Martin Rheinländer (Author), 2007, Analysis of Lattice-Boltzmann Methods, Munich, GRIN Verlag, https://www.grin.com/document/79991
