Two-stage Fourth-order Gas Kinetic Solver-based Compact Subcell Finite Volume Method for Compressible Flows over Triangular Meshes

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.