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.
Inhaltsverzeichnis (Table of Contents)
- Background and Outline
- Chapter 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
- Chapter 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
- Chapter 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
- Chapter 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
- Chapter 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
- Chapter 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
- Chapter 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
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
The main objective of this dissertation is to enhance the understanding of lattice-Boltzmann methods, particularly focusing on numerical phenomena such as initial and boundary layers. The work aims to achieve this through rigorous mathematical analysis of simplified model problems. Key themes include: * Analysis of lattice-Boltzmann methods using regular and irregular asymptotic expansions. * Investigation of singular limits in scaled Boltzmann-type equations. * Examination of consistency and stability of lattice-Boltzmann algorithms. * Exploration of numerical phenomena such as initial and boundary layers. * Development and application of analytical techniques to understand the behavior of lattice-Boltzmann methods.Zusammenfassung der Kapitel (Chapter Summaries)
Chapter 1: Introduction to lattice Boltzmann methods and their analysis: This chapter provides foundational knowledge of kinetic theory and lattice Boltzmann methods. It establishes the connection between the Boltzmann equation and macroscopic fluid dynamics, introduces different approaches to lattice Boltzmann methods, and lays out the theoretical framework for the analysis of these methods. The chapter also highlights the relationship between two-dimensional and one-dimensional lattice Boltzmann schemes, laying the groundwork for the simplified model problems used in subsequent chapters. The chapter concludes by introducing the concepts of consistency and stability within the context of numerical and asymptotic analysis.
Chapter 2: Scalings and singular limits on the basis of the D1P2 model: This chapter analyzes the D1P2 lattice-Boltzmann equation under hyperbolic and parabolic scalings. It examines the singular limits associated with each scaling and demonstrates convergence using a Fourier series approach. The chapter investigates the appearance of initial layers through multiscale expansions, providing detailed insights into the structure and coupling of the initial layer with the regular solution.
Chapter 3: Analysis of a D1P3 lattice-Boltzmann equation: This chapter extends the analysis to a D1P3 lattice-Boltzmann equation, proving convergence through an energy estimate and regular asymptotic expansions. It further investigates the limitations of regular expansions and introduces irregular expansions to handle arbitrary initial values, explicitly describing initial layers. The chapter concludes with a brief discussion of non-periodic boundary conditions.
Chapter 4: Consistency of a D1P3 lattice-Boltzmann algorithm: This chapter focuses on the consistency analysis of a D1P3 lattice-Boltzmann algorithm, employing a formal expansion to reveal structural insights. It defines and constructs consistent truncated expansions, establishing the connection between the algorithm and its macroscopic target equation. The chapter concludes with an examination of numerical initial layers.
Chapter 5: Long-term behavior of an advective lattice-Boltzmann scheme: This chapter investigates the long-term behavior of a hyperbolically scaled D1P2 algorithm, highlighting discrepancies between regular expansion predictions and actual numerical behavior. Numerical experiments reveal the presence of multiple timescales. Multiscale expansions are then used to provide a more accurate description of the algorithm's behavior, validating the analysis through numerical examples. The chapter concludes with a discussion of the relationship between formal stability analysis based on the multiscale expansion and the actual stability properties of the algorithm.
Chapter 6: Stability investigations around the D1P2 model: This chapter delves into the stability of the D1P2 algorithm. It establishes ℓ∞-stability for a parabolically scaled algorithm and performs a detailed spectral analysis, comparing eigenvalues with those of the advection-diffusion equation. It also explores the role of the CFL-condition, proving its necessity and sufficiency for stability in the case of a hyperbolically scaled algorithm. The chapter compares formal stability arguments from multiscale expansions with the actual stability behavior and discusses stability for the algorithm with bounce-back boundary conditions. Finally, the chapter explores potential generalizations of these results to the D1P3 algorithm.
Chapter 7: Asymptotic analysis of a numeric boundary layer: This chapter analyzes the emergence of numerical boundary layers. It examines a finite difference discretization of the one-dimensional Poisson equation with Dirichlet boundary conditions and analyzes the impact of different extrapolation schemes. The chapter proves stability and introduces the damping property of discrete inverse operators, which is used to derive an asymptotic expansion of the numerical error and finally to determine the convergence rate of the scheme. The chapter concludes with numerical experiments illustrating the key findings.
Schlüsselwörter (Keywords)
Lattice-Boltzmann methods, asymptotic analysis, numerical analysis, consistency, stability, convergence, initial layers, boundary layers, multiscale expansions, singular perturbations, advection-diffusion equation, Poisson equation, finite difference methods, spectral analysis, CFL condition, bounce-back boundary conditions.
Frequently Asked Questions: Analysis of Lattice-Boltzmann Methods
What is the main topic of this dissertation?
This dissertation focuses on enhancing the understanding of lattice-Boltzmann methods, particularly concerning numerical phenomena like initial and boundary layers. It achieves this through rigorous mathematical analysis of simplified model problems.
What are the key themes explored in the dissertation?
Key themes include the analysis of lattice-Boltzmann methods using regular and irregular asymptotic expansions; investigation of singular limits in scaled Boltzmann-type equations; examination of consistency and stability of lattice-Boltzmann algorithms; exploration of numerical phenomena such as initial and boundary layers; and the development and application of analytical techniques to understand the behavior of lattice-Boltzmann methods.
What methods are used to analyze the lattice-Boltzmann methods?
The dissertation employs various analytical techniques, including regular and irregular asymptotic expansions, multiscale expansions, Fourier series analysis, spectral analysis, and energy estimates to investigate the consistency, stability, and convergence of different lattice-Boltzmann schemes.
Which specific lattice-Boltzmann models are analyzed?
The dissertation primarily analyzes simplified one-dimensional models, specifically the D1P2 and D1P3 lattice-Boltzmann equations. These models allow for a detailed mathematical analysis of key numerical phenomena.
What are the main findings regarding the D1P2 model?
The analysis of the D1P2 model explores hyperbolic and parabolic scalings, investigating singular limits and the appearance of initial layers through multiscale expansions. Stability analysis is conducted, including examination of the CFL condition and spectral analysis of evolution matrices.
What are the main findings regarding the D1P3 model?
The D1P3 model is analyzed for consistency and convergence using regular and irregular asymptotic expansions, addressing the impact of initial conditions. The analysis considers both periodic and non-periodic boundary conditions.
How are initial and boundary layers addressed in the dissertation?
Initial and boundary layers are investigated through multiscale expansions, identifying different timescales and exploring the behavior of numerical solutions near boundaries. The impact of different boundary condition discretizations is also analyzed, particularly in the context of a finite difference discretization of the one-dimensional Poisson equation.
What is the significance of the stability analysis?
The stability analysis examines the conditions under which the numerical schemes remain stable. This includes exploring ℓ∞ and ℓ2 stability, the role of the CFL condition, and the spectral analysis of evolution matrices for different boundary conditions.
What is the overall conclusion of the dissertation?
The dissertation provides a comprehensive mathematical analysis of simplified lattice-Boltzmann models, improving the understanding of their behavior and numerical properties. It highlights the importance of considering various scaling regimes, the presence of multiple timescales, and the impact of boundary conditions on the accuracy and stability of the numerical solutions.
What are the key words associated with this dissertation?
Lattice-Boltzmann methods, asymptotic analysis, numerical analysis, consistency, stability, convergence, initial layers, boundary layers, multiscale expansions, singular perturbations, advection-diffusion equation, Poisson equation, finite difference methods, spectral analysis, CFL condition, bounce-back boundary conditions.
- Arbeit zitieren
- Martin Rheinländer (Autor:in), 2007, Analysis of Lattice-Boltzmann Methods, München, GRIN Verlag, https://www.hausarbeiten.de/document/79991
