Entropic Lattice Boltzmann Method

Abstract

Lattice Boltzmann methods are kinetic descriptions of fluid flow that are efficiently implemented through a stream and collide approach. The collision operation is typically an approximation of the microscopic physics, with the BGK linear approximation a widely used choice. It has a wide range of application in computational fluid dynamics. The honor thesis starts with an introduction to the kinetic theory of gas. First, we introduce the Liouville Equation that describes the evolution of particle distribution function of a system in phase space. With an analysis of the Liouville Equation, BBGKY hierarchy is introduced for the reduced distribution evolution. Using the Bogoliubov Hypothesis, we are able to close the hierarchy equation and derive the Boltzmann's Equation for kinetic theory of gas. An analysis of Boltzmann's Equation is presented, with the center of analysis being the Chapman-Enskog analysis that reproduces the Navier-Stokes Equation in its expansion. Then, the Lattice Boltzmann Method is introduced by describing collision and streaming steps. The formulation of Lattice Boltzmann Method is centered around discretizing the velocity space by using Gauss-Hermite Quadrature that gives a truncated version of the Maxwell-Boltzmann distribution. Under this truncation, Lattice Boltzmann Method becomes essentially a forward Euler scheme with BGK operator under a simplification of collision physics that is described by a limited direction of possible microscopic velocities. The BGK relaxation approximation however suffers from known deficiencies leading to instability at high Reynolds numbers that are the result of not satisfying a microscopic H-theorem. Entropic lattice methods seek an alternative relaxation procedure that is guaranteed to be unconditionally stable. They however typically are computationally expensive requiring solution of a nonlinear equation at each lattice site and each time step. This honor thesis further introduces the possibility of an alternative relaxation approach based upon geodesic transport on a statistical manifold within the overall framework of information geometry.