A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured mesh

Abstract

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.