Theoretical Analysis And Boundary Treatments Of The Lattice Boltzmann Method

The lattice Boltzmann method(LBM)has been a popular method for com-putational fluid dynamics.However,there are still a few open theoretical issues within the LBM that need to be addressed.This thesis is devoted to studying the theory including the stability,the corresponding macroscopic equations and boundary treatments of the LBM.First,by checking the Onsager-like relation proposed by Yong,we use the model with nine discrete velocities on a two-dimensional square lattice as an example to find a rigorous sufficient condition for the stability and convergence of the widely used multiple-relaxation-time model.Second,we apply the Maxwell iteration to the derivation of the macroscopic equations from the lattice Boltzmann equation(LBE).The approach uses the spatial(or time)step as the small expansion parameter instead of introducing the Knudsen number.Based on the derivation with convective scaling,we obtain the relation between the Mach number Ma and the time step ?t:Ma =O(?t1/3)for the LBE containing equilibrium distribution expanded with low Mach number and truncated to the third order.Third,based on the Maxwell iteration,we construct a class of single-node second-order schemes for the Dirichlet boundary conditions of the Navier-Stokes equations.With numerical tests,the second-order accuracy of two specific schemes are examined for both straight and curved boundaries.In addition,we show that the boundary scheme with a convex combination of distribution functions has good stability.We also use the Maxwell iteration to construct a scheme for the Neumann boundary conditions of the Cahn-Hilliard equation and simulate the moving contact line problems on curved boundaries.This work sets the analysis of the LBM on solid mathematical footing,thus making the theory more complete.