A third-order gas-kinetic scheme (GKS) based on the subcell finite volume (SCFV) method is developed for the Euler and Navier-Stokes equations on triangular meshes, in which a computational cell is subdivided into four subcells. The scheme combines the compact high-order reconstruction of the SCFV method with the high-order flux evolution of the gas-kinetic solver. Different from the original SCFV method using weighted least-square reconstruction along with a conservation correction, a constrained least-square reconstruction is adopted in SCFV-GKS so that the correction is not necessary and continuous polynomials inside each main cell can be reconstructed. For flows with large gradients, a compact hierarchical WENO limiting is adopted. In the gas-kinetic flux solver, the inviscid and viscous flux are coupled and computed simultaneously. Moreover, the spatial and temporal evolution are coupled nonlinearly which enables the developed scheme to achieve third-order accuracy in both space and time within a single stage. As a result, no quadrature point is required for the flux evaluation and the multi-stage Runge-Kutta method is unnecessary as well. A third-order k-exact finite volume GKS is also constructed with the help of k-exact reconstruction. Compared to the k-exact GKS, SCFV-GKS is compact, and with less computational cost for the reconstruction which is based on the main cell. Besides, a continuous reconstruction can be adopted in SCFV-GKS among subcells in smooth flow regions, which results in less numerical dissipation and less computational cost for flux evolution. Thus the efficiency of SCFV-GKS is much higher. Numerical tests demonstrate the strong robustness, high accuracy and efficiency of SCFV-GKS in a wide range of flow problems from nearly incompressible to supersonic flows with strong shock waves.

针对三角网格上的Euler和Navier-Stokes方程，开发了基于子单元有限体积（SCFV）方法的三阶气体动力学方案（GKS），其中将计算单元细分为四个子单元。该方案将SCFV方法的紧凑型高阶重构与气体动力学求解器的高阶通量演化结合在一起。与原始的使用加权最小二乘重建和保留校正的SCFV方法不同，在SCFV-GKS中采用了受约束的最小二乘重建，因此不需要校正，并且可以重建每个主像元内的连续多项式。对于具有大梯度的流，采用紧凑的分层WENO限制。在气体动力学通量求解器中，不粘流体通量和粘性通量被耦合并同时计算。而且，时空演化是非线性耦合的，这使开发的方案能够在单个阶段内在空间和时间上实现三阶精度。结果，通量评估不需要正交点，并且也不需要多级Runge-Kutta方法。还借助k精确重建构造了三阶k精确有限体积GKS。与k精确GKS相比，SCFV-GKS紧凑，并且基于主小区的重建计算成本较低。此外，在SCFV-GKS中可以在平滑流动区域的子电池之间采用连续重建，从而减少了数值耗散，并减少了通量演化的计算成本。因此，SCFV-GKS的效率要高得多。数值测试表明其强大的鲁棒性，