With discretized particle velocity space, a unified gas-kinetic scheme for entire Knudsen number flows has been constructed based on the Bhatnagar-Gross-Krook (BGK) model [J. Comput. Phys. 229 (2010), pp. 7747-7764]. In comparison with many existing kinetic schemes for the Boltzmann equation, besides accurate capturing of non-equilibrium flows, the unified method has no difficulty to get accurate solution in the continuum flow regime as well, such as the solution of the Navier-Stokes (NS)

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... ons. More importantly, in the continuum flow regime the time step used by the unified scheme is determined by the CFL condition, which can be many orders larger than the particle collision time. In some sense, the unified method overcomes the time step barrier for many kinetic methods, such as DSMC, direct Boltzmann solver, and many other kinetic solvers. The unified scheme is a multiscale method, where the macroscopic flow variables and microscopic gas distribution function are updated simultaneously. In an early approach in the Unified-BGK scheme, the heat flux in the BGK model is modified through the update of macroscopic flow variables, then this modification feeds back into the update of non-equilibrium gas distribution function. In this paper, we are going to develop a unified scheme for the Shakhov model, the so-called U-Shk, where the heat flux in corrected directly through the modification of gas distribution function. Theoretically, it will be shown that current U-Shk is more consistent than the U-BGK for the highly non-equilibrium flow computations. The study of shock structures from low to high Mach numbers will be presented and the simulation results will be compared with DSMC solutions as well as possible experimental measurements. The result improvement of U-Shk over U-BGK is clearly achieved. Based on the simulation results, now we fully believe that the unified scheme is an accurate and efficient flow solver in all Knudsen number flow regime. The classification of the various flow regimes based on the dimensionless parameter, the Knudsen number, is a measure of the degree of rarefaction of the medium. The Knudsen number Kn is defined as the ratio of the mean free path to a characteristic length scale of the system. In the continuum flow regime where Kn < 0.001, the Navier-Stokes equations with linear relations between stress and strain and the Fourier's law for heat conduction are adequate to model the fluid behavior. For flows in the continuum-transition regime (0.1 < Kn < 1), the Navier-Stokes equations are known to be inadequate. This regime is important for many practical engineering problems, such as the simulation of microscale flows and hypersonic flow around space vehicles in low earth orbit. Hence, there is a strong desire and requirement for accurate models which give reliable solutions with lower computational costs. The Boltzmann equation describes the flow in all flow regimes; continuum, continuumtransition and free molecular. However, to design an accurate and efficient Boltzmann solver for both rarefied and continuum regimes seem very challenge. One of the outstanding numerical techniques available for solving the Boltzmann equation is the direct simulation Monte Carlo (DSMC) [5] method. The DSMC method is a widely used technique in the numerical prediction of low density flows. However, in the continuumtransition regime, where the density is not low enough, the DSMC requires a large number of particles for accurate simulation, which makes the technique expensive both in terms of the computation time and memory requirement. At present, the accurate modeling of realistic configurations, such as aerospace vehicles in three dimensions by the DSMC method for Kn << 1, is beyond the currently available computing power. The DSMC method requires that the time step and cell size are less than the particle collision time and mean free path, which subsequently introduce enormous computational cost in the high density regime. The Boltzmann equation is valid from the continuum flow regime to the free molecule flow. So, theoretically a direct Boltzmann solver which is valid in the whole range of Knudsen number can be developed if the numerical discretization is properly designed. In the framework of deterministic approximation, the most popular class of methods is based on the so-called discrete velocity methods (DVM) or Discrete Ordinate Method (DOM) of the Boltzmann equation [8, 26, 15, 14, 1, 16] . These methods use regular discretization of par-