Analysis of Lattice-Boltzmann Methods
GRIN Verlag, 2007 - 384 pages
Doctoral Thesis / Dissertation from the year 2007 in the subject Mathematics - Analysis, University of Constance (Fachbereich Mathematik & Statistik), 69 entries in the bibliography, language: English, comment: Die Arbeit wurde mit 1 (magna cum laude bewertet) und enthalt farbige Abbildungen., abstract: 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 par
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addend advection advection equation ansatz approximate asymptotic analysis asymptotic expansion asymptotic order functions asymptotic similarity behavior Boltzmann equation bounce-back characteristic polynomial collision operator computation consider constant convergence D1P3 model D1P3 scheme defined denotes depend derivative diffusion equation Dirichlet boundary conditions discrete velocity eigenvalues eigenvectors equilibrium estimate evolution equation evolution matrix finite Fourier series function f Furthermore grid spacing Hence IBVP initial condition initial layer initial value problem integral iteration kinetic lattice-Boltzmann algorithm lattice-Boltzmann equation lattice-Boltzmann methods lattice-Boltzmann scheme lemma linear macroscopic mass Navier-Stokes Navier-Stokes equation norm normed vector space numerical obtain parabolic scaling parameter particles periodic boundary conditions polynomial population function Proof proposition recursion regular expansion residual respect right hand side satisfies scalar second order solution spatial spectral limit set stability stencil target equation target problem term theorem triangle inequality vanish yields zero zeroth
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Page 176 - For linear systems, a difference scheme is convergent if and only if it is consistent and stable. Proof of this theorem can be found in Sod (1985) or Richtmyer and Morton (1967).
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Page 243 - Proof: The first part is an immediate consequence of theorem (6.2) since...
Page 231 - Since the characteristic polynomial is the same for a matrix and its transposed, we may restrict ourselves by (6.3) to the case of L. Writing out the determinant and expanding it by Leibniz...
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