Analysis of Lattice Boltzmann Boundary Conditions

In this dissertation, we investigate a class of standard linear and nonlinear lattice Boltzmann methods from the point of view of mathematical analysis. First we study the consistency of the lattice Boltzmann method on a bounded domain by means of asymptotic analysis. From the analysis of the lattice Boltzmann update rule, we find a representation of the lattice Boltzmann solutions in form of truncated regular expansions, which clearly exhibit the relation to solutions of the Navier-Stokes equation. Through the analysis of the initial conditions and the well-known bounce back boundary rule, we demonstrate the general procedure to integrate the boundary analysis process in the whole analysis, and find that our approach can reliably predict the accuracy of the lattice Boltzmann solutions as approximations to Navier-Stokes solutions. Next, a rigorous convergence proof is achieved for the class of standard linear and nonlinear lattice Boltzmann methods considered in this thesis. Concentrating on realizations of Dirichlet velocity boundary conditions, we then investigate the consistency of several existing implementations, predict their accuracy, and their advantages and shortcomings. In order to overcome a general drawback of the methods, we construct a class of purely local boundary treatments. All of these methods lead to a second order accurate velocity and a first order accurate pressure. A careful numerical comparison of their properties such as stability, mass conservation and error behavior is presented, as well as a guide for choosing a boundary implementation among the various possibilities. Regarding Navier-Stokes outflow conditions which are hardly studied in the lattice Boltzmann literature, we deal with three kind of Neumann-type conditions. We have proposed their implementations in the lattice Bolzmann framework, and briefly carry out their consistency analysis. Several numerical results demonstrate the capability of these outflow treatments. For the unsteady benchmark problem like flows around fixed cylinders in an infinitely long channel, the proposed do-nothing and zero normal stress conditions perform very well. For the steady flow, all of the methods produce convincing results.