A BGK model on the 2D adaptive unstructured mesh is developed within a finite volume framework, which consists of two steps:the solution reconstruction step and the gas evolution step. The computation domain is tessellated with triangular elements. In the reconstruction step, a high-order ENO scheme based on the least-square principle is employed to obtain the derivatives. In the gas evolution step, with the axis rotation, the flux through each side of the triangular element is obtained. Some test cases validate the capability of the current numerical approach to provide a highly accurate solution in complex geometries. It is observed that the BGK model provides a reasonable alternative to Riemann solvers.

It develops and describes a new method, the ink soak method, for adding boundary points in generation of three dimensional unstructured meshes with Delaunay method. In the Delaunay method, there is an essential defect, i.e., the generated meshes can cross with bound of domain and do not coincide with it, specially in 3D problems. To overcome this difficulty, some complicated methods to check and to correct them were designed. All these method need numerous computational time. This problem is greatly simplified by the present ink soak method. Some 3-D examples are given and demonstrate the efficiency of the ink soak method for unstructured grid generation in complex regions.