Abstract: The discontinuous Galerkin method is developed based on high-resolution FDM and FVM.They achieve great success for solving problems in computational fluid dynamics.The basic conseption and computational scheme of the method for numerical solution of conservation laws are introduced.The order analyses and influence of the limiters are discussed by simple numerical simulations.Meantime,the numerical results of problems in hydraulic dynamic,traffic flow problems and gas dynamic problems are given.Morever,the summary of the advances in numerical applications for elliptic problems,parabolic problems and convection diffusion problems,Hamilton-Jacobi and Navier-Stokes equations are presented.

Key words: discontinuous finite element method, high resoution, numerical simulations, advances in applications