  To meet the demand for complex geometries and high resolutions of small-scale
flow structures, a two-stage fourth-order subcell finite volume (SCFV) method
combining the gas-kinetic solver (GKS) with subcell techniques for compressible
flows over (unstructured) triangular meshes was developed to improve the
compactness and efficiency. Compared to the fourth-order GKS-based traditional
finite volume (FV) method, the proposed method realizes compactness effectively
by subdividing each cell into a set of subcells or control volumes (CVs) and
selecting only face-neighboring cells for high-order compact reconstruction.
Because a set of CVs share a solution polynomial, the reconstruction is more
efficient than that for traditional FV-GKS, where each CV needs to be
separately reconstructed. Unlike in the single-stage third-order SCFV-GKS, both
accuracy and efficiency are improved significantly by two-stage fourth-order
temporal discretization, for which only a second-order gas distribution
function is needed to simplify the construction of the flux function and reduce
computational costs. For viscous flows, it is not necessary to compute the
viscous term with GKS. Compared to the fourth-stage Runge--Kutta method, one
half of the stage is saved for achieving fourth-order time accuracy, which also
helps to improve the efficiency. Therefore, a new high-order method with
compactness, efficiency, and robustness is proposed by combining the SCFV
method with the two-stage gas-kinetic flux. Several benchmark cases were tested
to demonstrate the performance of the method in compressible flow simulations.

Numerical Analysis;Numerical Analysis;Computational Physics;Fluid Dynamics