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Numerical Analysis of Lattice Boltzmann Methods for the Heat Equation on a Bounded Interval 1st Edition by WeiB Jan Philipp ISBN 3866440693 9783866440692

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Numerical Analysis of Lattice Boltzmann Methods for the Heat Equation on a Bounded Interval Weiß Jan-Philipp Digital Instant Download

Author(s): Weiß Jan-Philipp
ISBN(s): 9783866440692, 3866440693
Edition: PhD Thesis
File Details: PDF, 9.59 MB
Year: 2006
Language: english
SKU: EB-999212 Category: Tag:

Numerical Analysis of Lattice Boltzmann Methods for the Heat Equation on a Bounded Interval 1st Edition by WeiB Jan Philipp – Ebook PDF Instant Download/Delivery: 3866440693, 9783866440692
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ISBN 10: 3866440693 
ISBN 13: 9783866440692
Author: WeiB Jan Philipp

Lattice Boltzmann methods are a promising approach for the numerical solution of fluid-dynamic problems. We consider the one-dimensional Goldstein-Taylor model with the aim to answer some of the questions concerning the numerical analysis of lattice Boltzmann schemes. Discretizations for the solution of the heat equation are presented for a selection of boundary conditions. Stability and convergence of the solutions are proved by employing energy estimates and explicit Fourier representations.

Numerical Analysis of Lattice Boltzmann Methods for the Heat Equation on a Bounded Interval 1st Table of contents:

Chapter 1: Introduction

  • 1.1 Overview of Lattice Boltzmann Methods (LBM)

  • 1.2 Heat Equation and Its Numerical Solutions

  • 1.3 Objectives of the Study

  • 1.4 Structure of the Book/Paper

Chapter 2: Theoretical Background

  • 2.1 The Heat Equation

    • 2.1.1 Formulation of the Heat Equation

    • 2.1.2 Classical Analytical Solutions to the Heat Equation

  • 2.2 Lattice Boltzmann Method (LBM) Overview

    • 2.2.1 The Boltzmann Equation and Lattice Gas Automata

    • 2.2.2 Discrete Velocity Models in Lattice Boltzmann Methods

  • 2.3 Numerical Methods for the Heat Equation

    • 2.3.1 Finite Difference Methods

    • 2.3.2 Finite Element Methods

    • 2.3.3 Lattice Boltzmann Methods in Heat Transfer

Chapter 3: Mathematical Formulation of the Problem

  • 3.1 Heat Equation on a Bounded Interval

    • 3.1.1 Boundary Conditions: Dirichlet and Neumann

    • 3.1.2 Initial Condition Setup

  • 3.2 Discretization of the Heat Equation

    • 3.2.1 Space Discretization and Grid Setup

    • 3.2.2 Time Discretization and Scheme Selection

  • 3.3 Introduction to Lattice Boltzmann Discretization for Heat Transfer

Chapter 4: Lattice Boltzmann Models for Heat Transfer

  • 4.1 D2Q9 Model and Its Application to Heat Transfer

  • 4.2 Implementation of Thermal Boundary Conditions in LBM

    • 4.2.1 Handling Dirichlet and Neumann Conditions in LBM

    • 4.2.2 Boundary Treatment Techniques in the LBM Framework

  • 4.3 Thermal Conductivity and Diffusivity in Lattice Boltzmann

    • 4.3.1 Thermal Diffusivity in LBM Simulations

    • 4.3.2 Numerical Stability and Accuracy in Heat Transfer Simulations

Chapter 5: Numerical Stability and Convergence Analysis

  • 5.1 Stability Analysis of Lattice Boltzmann Methods for Heat Transfer

    • 5.1.1 von Neumann Stability Analysis

    • 5.1.2 CFL Condition and Stability Criteria for LBM

  • 5.2 Error Analysis of Lattice Boltzmann Methods

    • 5.2.1 Global and Local Errors in Discretization

    • 5.2.2 Consistency and Convergence Results

  • 5.3 Convergence Rates for Heat Equation Simulations Using LBM

Chapter 6: Implementation of Lattice Boltzmann Methods

  • 6.1 Discrete Velocity Models and Computational Domain

    • 6.1.1 D2Q9 and Other Lattice Arrangements

    • 6.1.2 Grid Setup and Computational Considerations

  • 6.2 Code Implementation and Numerical Experiments

    • 6.2.1 Software and Programming Languages Used

    • 6.2.2 Parallelization and Performance Optimization

  • 6.3 Validation of Lattice Boltzmann Method for Heat Equation

    • 6.3.1 Comparison with Analytical Solutions

    • 6.3.2 Benchmarking Against Other Numerical Methods

Chapter 7: Numerical Results and Discussion

  • 7.1 Numerical Simulations for Various Boundary Conditions

    • 7.1.1 Dirichlet Boundary Conditions

    • 7.1.2 Neumann Boundary Conditions

  • 7.2 Impact of Grid Resolution on Accuracy and Stability

    • 7.2.1 Grid Refinement Study

    • 7.2.2 Convergence Behavior with Different Grid Sizes

  • 7.3 Time Step Analysis

    • 7.3.1 Optimal Time Step Selection for Stability and Accuracy

    • 7.3.2 Comparison with Other Numerical Methods

  • 7.4 Error Analysis and Convergence Behavior

    • 7.4.1 Discrepancy Between Numerical and Analytical Solutions

    • 7.4.2 Convergence Rates in Various Settings

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Tags: WeiB Jan Philipp, Numerical Analysis, Lattice Boltzmann