As an extension of a fourth-order compact gas kinetic scheme (GKS) on structured mesh [24], this work is about the development of a third-order compact GKS on unstructured mesh for the compressible Euler and Navier-Stokes solutions. Based on the time accurate high-order gas-kinetic evolution solution, the time-dependent gas distribution function at a cell interface in GKS provides not only the flux function and its time derivative, but also the time accurate flow variables there at the next time level. As a result, besides updating the conservative flow variables inside each control volume through the interface fluxes, the cell averaged first-order spatial derivatives of flow variables can be obtained as well using the updated flow variables at the closed cell interfaces around that cell through the divergence theorem. Therefore, with the cell-averaged flow variables and their first-order spatial derivatives inside each cell, the Hermite WENO (HWENO) techniques can be naturally implemented for the compact high-order reconstruction at the beginning of the next time step. Following the reconstruction technique in [64], a new HWENO reconstruction on triangular mesh is designed in the current scheme. Combined with the two-stage temporal discretization and second-order time accurate flux function, a third-order compact scheme on unstructured mesh has been constructed. Accurate solutions can be obtained for both inviscid and viscous flows without sensitive dependence on the quality of triangular mesh. The robustness and accuracy of the scheme have been validated through many cases, including strong shocks in the hypersonic viscous flow and smooth Navier-Stokes solution.

作为对结构化网格上的四阶压缩气体动力学方案（GKS）的扩展[24]，这项工作是针对非结构化网格上可压缩的Euler和Navier-Stokes解决方案的三阶紧凑型GKS的开发。基于时间精确的高阶气体动力学演化解决方案，GKS单元界面中随时间变化的气体分布函数不仅提供通量函数及其时间导数，还提供下一次的时间精确流量变量水平。结果，除了通过界面通量更新每个控制体积内的保守流动变量外，还可以使用通过该单元周围的封闭单元界面处的更新后的流动变量，获得流动变量的单元平均一阶空间导数。发散定理。因此，借助每个单元内部的单元平均流量变量及其一阶空间导数，Hermite WENO（HWENO）技术可以自然地在下一个时间步开始时实现紧凑的高阶重构。继[64]中的重建技术之后，在当前方案中设计了一种新的三角形网格HWENO重建。结合两阶段时间离散化和二阶时间精确通量函数，构造了非结构网格的三阶紧致格式。在不敏感地依赖于三角形网格的质量的情况下，对于粘性和粘性流都可以获得准确的解决方案。该方案的鲁棒性和准确性已在许多情况下得到了验证，包括高超声速粘性流中的强烈冲击和光滑的Navier-Stokes解决方案。