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)
- Preface
- Introduction
- The Lattice-Boltzmann Method
- The Basic Idea
- The Discrete Velocity Set and the Lattice
- The Collision Operator and the Equilibrium Distribution
- The BGK Approximation
- Consistency and Stability of the Lattice-Boltzmann Method
- Implementation of Boundary Conditions
- A Singularly Perturbed System
- The Mathematical Model
- Asymptotic Analysis
- Derivation of the Reduced Problem
- The Structure of the Inner Solution
- Numerical Investigations
- The Finite Difference Scheme
- The Lattice-Boltzmann Method
- Convergence Analysis
- Numerical Results
- Summary and Outlook
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This dissertation aims to investigate the applicability of the Lattice-Boltzmann method for solving a singularly perturbed system. The work focuses on both asymptotic analysis and numerical simulations to study the behavior of the system.
- Application of the Lattice-Boltzmann method to a specific singularly perturbed problem
- Asymptotic analysis of the system to derive the reduced problem and understand the behavior of the solution
- Development of a numerical scheme based on finite differences for comparison with the Lattice-Boltzmann method
- Numerical investigations and convergence analysis to validate the results obtained by both methods
- Exploring the efficiency and accuracy of the Lattice-Boltzmann method for solving this type of problem
Zusammenfassung der Kapitel (Chapter Summaries)
- Preface: The dissertation originates from the BMBF project "Adaptive Grid Control for Lattice-Boltzmann Methods for the Simulation of Filling Processes in the Foundry Area" at the Fraunhofer Institute for Techno- and Wirtschaftsmathematik (ITWM) in Kaiserslautern. This section expresses gratitude to the dissertation advisor and co-advisor for their support and guidance throughout the research process.
- Introduction: This chapter provides an overview of the problem under investigation, including a brief introduction to singularly perturbed systems and the Lattice-Boltzmann method. It also outlines the objectives and structure of the dissertation.
- The Lattice-Boltzmann Method: This chapter presents a comprehensive description of the Lattice-Boltzmann method, including its basic principles, the discrete velocity set and lattice, the collision operator, the BGK approximation, consistency and stability, and implementation of boundary conditions.
- A Singularly Perturbed System: This chapter focuses on the mathematical model of the singularly perturbed system, including its derivation and analysis. It examines the asymptotic behavior of the system and derives the reduced problem.
- Numerical Investigations: This chapter discusses the numerical methods used for solving the singularly perturbed system. It presents both the finite difference scheme and the Lattice-Boltzmann method, followed by an analysis of their convergence properties and a comparison of their numerical results.
Schlüsselwörter (Keywords)
The dissertation focuses on the application of the Lattice-Boltzmann method to a singularly perturbed system, exploring its efficiency and accuracy compared to a traditional finite difference scheme. Key topics include asymptotic analysis, numerical simulation, and convergence analysis within the context of Lattice-Boltzmann methods and singularly perturbed problems.
- Quote paper
- Martin Rheinländer (Author), 2007, Analysis of Lattice-Boltzmann Methods, Munich, GRIN Verlag, https://www.grin.com/document/79991
