High order numerical methods have been widely studied in computational fluid dy-namics in the past20to30years, which have high order, high resolution, low dissipationproperties, such properites are suitable for vortex dominated flow, turbulence, aeroa-coustics and many viscosity dominated flows. The discontinuous Galerkin method is aclass of finite element method using discontinuous piecewise polynomial space for thenumerical solution and the test function spaces. It is usually coupled with a high ordertime discretization method. The method becomes very popular in computational fluiddynamics and other areas of applications. It combines features of finite element meth-ods and finite volume methods, and they are suitable for hyperbolic conservation lawsand convection dominated problems. In this dissertation, there are three parts of work:In chapter2, we propose a Runge-Kutta （RK） central discontinuous Galerkin （CDG）gas-kinetic BGK method for the Navier-Stokes equations. The proposed method isbased on the CDG method defined on two sets of overlapping meshes to avoid discon-tinuous solutions at cell interfaces, as well as the gas-kinetic BGK model to evaluatefluxes for both convection and difusion terms. Redundant representation of the numer-ical solution in the CDG method ofers great convenience in the design of gas-kineticBGK fluxes. Specifically, the evaluation of fluxes at cell interfaces of one set of compu-tational mesh is right inside the cells of the staggered mesh, hence the correspondingparticle distribution function for flux evaluation is much simpler than that in existinggas-kinetic BGK methods. As a central scheme, the proposed CDG-BGK has doubledthe memory requirement as the corresponding DG scheme; on the other hand, for theconvection part, the CFL time step constraint of the CDG method for numerical sta-bility is relatively large compared with that for the DG method. Numerical boundaryconditions have to be treated with special care. Numerical examples for1D and2Dviscous flow simulations are presented to validate the accuracy and robustness of theproposed RK CDG-BGK method.In chapter3, we propose a local discontinuous Galerkin （LDG） kinetic flux vectorsplitting （KFVS） method for the Navier-Stokes equations. The proposed method isbased on the LDG formulation to rewrite Navier-Stokes equations as a first order sys- tem, and uses KFVS scheme to evaluate fluxes at cell interfaces for both convectionand difusion terms. The particle distribution function is split to equilibrium and non-equilibrium parts by the micro-macro decomposition method, where convection anddifusion terms can be obtained by moments of the equilibrium and non-equilibriumparts respectively. Such design of numerical fluxes at cell interfaces are more physicallyrelevant than the traditional LDG method with alternating or central fluxes. Com-pared with the gas kinetic BGK method, the time evolution is realized by a high orderTVD RK time discretization. Hence no explicit reconstruction of the equilibrium statealong characteristics, which is algebraically complicated and computational involved,is needed. Stability analysis for a linear convection difusion equation demonstratesthe L2stability of the proposed scheme. Numerical examples for1D and2D viscousflow simulations are presented to validate the accuracy and robustness of the proposedLDG-KFVS method.In chapter4, we propose a bound-preserving Runge-Kutta discontinuous Galerkin（RKDG） method as an efcient, efective and compact numerical approach for numericalsimulations of trafc flow problems on networks, with arbitrary high order accuracy.Road networks are modeled by graphs, composed of a finite number of roads that meetat junctions. On each road, a scalar conservation law describes the dynamics, whilecoupling conditions are specified at junctions to define flow separation or convergenceat the points where roads meet. We incorporate such coupling conditions in the RKDGframework, and apply an arbitrary high order bound preserving limiter to the RKDGmethod to preserve the physical bounds on the network solutions. We showcase theproposed algorithm on several benchmark test cases from the literature, as well asseveral new challenging examples with rich solution structures. Modeling and simulationof Cauchy problems for trafc flows on networks is notorious for lack of uniqueness or（Lipschitz） continuous dependence. The discontinuous Galerkin method proposed heredeals elegantly with these problems, and is perhaps the only realistic and efcient high-order method for network problems.