Vector-BGK Models Based Numerical Methods And Boundary Treatments For The Incompressible Navier-Stokes Equations

The incompressible Navier-Stokes equations have a wide range of applications,and solv-ing them numerically has drawn much attention,especially for the flow problems with complex geometries.We concentrate on a class of vector-BGK models for the incompressible Navier-Stokes equations from the PDE's viewpoints and the numerical methods in this thesis.More-over,we also focus on the corresponding boundary treatments with the accuracy and stability analysis.This thesis consists of the following five parts.Firstly,from the PDE's viewpoints,we prove that a class of discrete-velocity vector-BGK models can be used as approximations to the multi-dimensional incompressible Navier-Stokes equations.Simple and concise stability conditions are obtained for a number of concrete 2-or 3-dimensional BGK models.Secondly,using the Maxwell iteration,we analyze the accuracy of the lattice Boltzmann method based on the discrete-velocity vector-BGK models.Moreover,we prove the corre-sponding numerical stability.Thirdly,in order to deal with the Dirichlet boundary condition,we propose a vector-type bounce-back boundary scheme.The scheme is shown to have second-order accuracy if the boundary is located at the middle of two neighboring grid points.Furthermore,we base on the new scheme and construct a parameterized second-order boundary scheme with accuracy independent of the location of the boundary.We also show the numerical stability of these schemes.Fourthly,we study a new class of numerical method based on the vector-BGK models(also called the lattice Boltzmann method with general propagation).The incompressible Navier-Stokes equations can be derived from it by using the Maxwell iteration.We then propose a new boundary scheme to accompany the vectorial lattice Boltzmann method with general propagation.We also show that the boundary scheme has second-order accuracy when the boundary is located at the middle of two neighboring lattice nodes.Moreover,inspired by the novel boundary scheme,we utilize an auxiliary boundary scheme to construct a family of parameterized second-order boundary schemes with accuracy independent of the location of boundary.Additionally,we also show the stability of these schemes.Finally,we study a three-dimensional seven-velocity(D3N7)vectorial model including various types of approximations to the pressure and propose a family of two-parameterized second-order boundary schemes with accuracy independent of the boundary location.We also show the numerical stability of the vectorial model including nonlinear approximations to the pressure.