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SUMMARYBased on the Bhatnagar-Gross-Krook (BGK) Boltzmann model equation, the uniÿed simpliÿed velocity distribution function equation adapted to various ow regimes can be presented. The reduced velocity distribution functions and the discrete velocity ordinate method are developed and applied to remove the velocity space dependency of the distribution function, and then the distribution function equations will be cast into hyperbolic conservation laws form with non-linear source terms. Based on the unsteady time-splitting technique and the non-oscillatory, containing no free parameters, and dissipative (NND) ÿnite-di erence method, the gas kinetic ÿnite-di erence second-order scheme is constructed for the computation of the discrete velocity distribution functions. The discrete velocity numerical quadrature methods are developed to evaluate the macroscopic ow parameters at each point in the physical space. As a result, a uniÿed simpliÿed gas kinetic algorithm for the gas dynamical problems from various ow regimes is developed. To test the reliability of the present numerical method, the one-dimensional shocktube problems and the ows past two-dimensional circular cylinder with various Knudsen numbers are simulated. The computations of the related ows indicate that both high resolution of the ow ÿelds and good qualitative agreement with the theoretical, DSMC and experimental results can be obtained.

Abstract. The method of mapping function was first proposed by Henrick et al. [J. Comput. Phys. 207:542-547 (2005)] to adjust nonlinear weights in [0,1] for the fifth order WENO scheme, and through which the requirement of convergence order is satisfied and the performance of the scheme is improved. Different from Henrick’s method, a concept of piecewise polynomial function is proposed in this study and corresponding WENO schemes are obtained. The advantage of the new method is that the function can have a gentle profile at the location of the linear weight (or the mapped nonlinear weight can be close to its linear counterpart), and therefore is favorable for the resolution enhancement. Besides, the function also has the flexibility of quick convergence to identity mapping near two endpoints of [0,1], which is favorable for improved numerical stability. The fourth-, fifth- and sixth-order polynomial functions are constructed correspondingly with different emphasis on aforementioned flatness and convergence. Among them, the fifth-order version has the flattest profile. To check the performance of the methods, the 1-D Shu-Osher problem, the 2-D Riemann problem and the double Mach reflection are tested with the comparison of WENO-M, WENO-Z and WENO-NS. The proposed new methods show the best resolution for describing shear-layer instability of the Riemann problem, and they also indicate high resolution in computations of double Mach reflection, where only these proposed schemes successfully resolved the vortex-pairing phenomenon. Other investigations have shown that the single polynomial mapping function has no advantage over the proposed piecewise one, and it is of no evident benefit to use the proposed method for the symmetric fifth-order WENO. Overall, the fifth-order piecewise polynomial and corresponding WENO scheme are suggested for resolution improvement.

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