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In this paper an even higher-order compact GKS up to sixth order of accuracywill be constructed for the shock and acoustic wave computation on unstructuredmesh. The compactness is defined by the physical domain of dependence for anunstructured triangular cell, which may involve the closest neighbors ofneighboring cells. The compactness and high-order accuracy of the scheme arecoming from the consistency between the high-order initial reconstruction andthe high-order gas evolution model under GKS framework. The high-orderevolution solution at a cell interface provides not only a time-accurate fluxfunction, but also the time-evolving flow variables. Therefore, thecell-averaged flow variables and their gradients can be explicitly updated atthe next time level from the moments of the same time-dependent gasdistribution function. Based on the cell averages and cell-averagedderivatives, both linear and nonlinear high-order reconstruction can beobtained for macroscopic flow variables in the evaluation of local equilibriumand non-equilibrium states. The current nonlinear reconstruction is acombination of WENO and ENO methodology. The initial piecewise discontinuousreconstruction is used for the determination non-equilibrium state and anevolved smooth reconstruction for the equilibrium state. The evolution model ingas-kinetic scheme is based on a relaxation process from non-equilibrium toequilibrium state. The accuracy, efficiency, and robustness of the scheme havebeen validated. The main conclusion of the paper is that beyond the first-orderRiemann solver, the use of high-order gas evolution model seems necessary inthe development of high-order schemes.

A compact high-order gas-kinetic scheme on unstructured mesh for acoustic and shock wave computations