This paper presents an efficient high-order finite volume method for solution of compressible Navier-Stokes equations on unstructured grids. In this method, a high-order polynomial which is based on Taylor series expansion is applied to approximate the solution function within each control cell. The derivatives in the Taylor series expansion are approximated by the functional values at the cell centers of the considered cell and its neighboring cells using the mesh-free least square-based finite difference (LSFD) scheme. Naturally, this least square-based finite difference-finite volume (LSFD-FV) method inherits appealing characteristics of the LSFD scheme, i.e., the simple algorithm and easy implementation. In addition, since the LSFD scheme is mesh-free in nature, the developed high-order LSFD-FV method is endowed with the flexibility to handle the multi-dimensional problems with complex geometries on arbitrary grids. Different from other high-order methods, this LSFD-FV method applies a novel and effective strategy, i.e., the discrete gas-kinetic flux solver (DGKFS), to compute numerical fluxes at the cell interface. In this way, the inviscid and viscous fluxes are simultaneously and effectively evaluated by local reconstruction of solutions for the Boltzmann equation. The efficient time marching strategy coupled with the high-order LSFD-FV method is adopted to solve the resultant ordinary differential equations with the matching accuracy. A series of numerical examples are presented to validate the accuracy, robustness and flexibility of the developed method on unstructured grids. Numerical results demonstrate the superior performance of the developed high-order method on simulating compressible viscous flows, in comparison with the low-order counterpart and the k-exact method of the same order of accuracy.

本文提出了一种有效的高阶有限体积方法，用于求解非结构化网格上的可压缩Navier-Stokes方程。在这种方法中，基于泰勒级数展开的高阶多项式被应用于近似每个控制单元内的解函数。泰勒级数展开式中的导数使用无网格的最小二乘有限差分（LSFD）方案，通过考虑的单元及其相邻单元的单元中心处的函数值进行近似。自然地，这种基于最小二乘的有限差分-有限体积（LSFD-FV）方法继承了LSFD方案的吸引人的特征，即简单的算法和易于实现的方法。另外，由于LSFD方案本质上是无网格的，所开发的高阶LSFD-FV方法具有灵活性，可以处理任意网格上具有复杂几何形状的多维问题。与其他高阶方法不同，此LSFD-FV方法应用了一种新颖有效的策略（即离散气体动力学通量求解器（DGKFS））来计算单元界面处的数值通量。通过这种方式，通过对Boltzmann方程解的局部重建，可以同时有效地评估不粘稠通量和粘性通量。采用高效的时间行进策略结合高阶LSFD-FV方法，以匹配的精度求解所得的常微分方程。给出了一系列数值例子，以验证所开发方法在非结构化网格上的准确性，鲁棒性和灵活性。精确度相同的k-精确方法。