We study numerical integration over arbitrary interface and arbitrary domain on block-structured adaptive mesh.Arbitrary interface and arbitrary domain are described by a level set function.We first describe numerical methods on uniform Cartesian grid.Then we extend the methods to block-structured adaptive mesh.Numerical calculations demonstrate that adaptive mesh methods are second-order accurate.Compared with uniform mesh methods, the adaptive mesh methods reduce needs on computer storage significantly.

A numerical scheme based on lattice Boltzman method(LBM) is presented to deal with curved boundaries with second-order accuracy. The scheme is applied to steady laminar flow within a quarter-circular cavity with curved boundaries and proved to be relatively reliable, accurate and feasible. Streamlines and vorticity contours are obtained to analyse the variation of flow field with Reynolds number. The radian between verticity centre of clockwise flow and x-axis decreases with increase of Re from 10 to 100, while the radian between verticity centre of anticlockwise flow and the x-axis shows inverse tendency.

The kinetic flux vector splitting method is combined with the two-time-step algorithm to simulate the inviscid ideal gas flow. The flux vectors of the equation are split on the basis of the local equilibrium Maxwell-Boltzmann distribution. Three 1-D standard test cases including shock waves, contact discontinuities and rarefaction waves are presented and compared with the exact solution.

A new lattice Boltzmann method is proposed for constructing a 1-D scheme by using bi distribution functions and Godunov scheme of decomposition. The Ideal Case(IC) problem has been hence removed.

Based on the equivalence between the aerdynamical Euler equation and the shallow water long wave equation, the equilibrum distribution function of the orginal LBM is improved, and hence the macrodynamical equation of shallow water long waves is derived. The resoults of one dimentional numerical test are compaired with those obtained by discrete velocity model.

A new lattice Boltzmann model is proposed for treating Euler equations. The numerical example shows that it can be used to simulate shock wave and contact discontinuity. The results are comparable with those obtained by traditional methods.

A Lattice Gas Method has been used for conservation equation in a model equation by defined general distribution function. The model became a bridge between Lattice Gas Method and Finite Difference Method. It is shown that the method is effective by simulating model equation with MacCormark and Warming Beam schemes.

The 7-Bit model of Lattice Boltzmann Method is used for simulation the porous media. Darcy law are found and the relationships of void ratio coefficient of viscosity and density with permeabilities are obtained as well.

We discuss a alternative technique to the lattice gas automata for the study of viscous flows. It is shown that the lattice gas automata with real number population deriving from the HPP model being free from microscopic fluctuations. provides a new appealing tool to simulate viscous flows. Numerical results pertaining to a two-dimensional flow past a square cylinder are reported and compared with experimental data available in the literature.

A approach of the calculation for the regular Coulomb wave function Ft (η,ρ) (η<0)and its main features depended on η,l and ρ are described. A example about their applications to integral calculations in electron-ion collision is given.