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In this paper, a class of compact higher-order gas-kinetic schemes (GKS) withspectral resolution will be presented. Based on the high-order gas evolutionmodel in GKS, both the interface flux function and conservative flow variablescan be evaluated explicitly from the time-accurate gas distribution function.As a result, inside each control volume both the cell-averaged flow variablesand their cell-averaged gradients can be updated within each time step. Theflow variable update and slope update are coming from the same physicalsolution at the cell interface. Different from many other approaches, such asHWENO, there are no additional governing equations in GKS for the slopes orequivalent degrees of freedom independently inside each cell. For nonlinear gasdynamic evolution, the above compact linear reconstruction from the symmetricstencil can be divided into sub-stencils and apply a biased nonlinear WENO-Zreconstruction. In GKS, the time evolution solution of the gas distributionfunction at a cell interface covers a physical process from an initialnon-equilibrium state to a final equilibrium one. The GKS evolution modelsunifies the nonlinear and linear reconstructions in gas evolution process forthe determination of a time-dependent gas distribution function. Thisdynamically adaptive model helps to solve a long lasting problem in thedevelopment of high-order schemes about the choices of the linear and nonlinearreconstructions. Compared with discontinuous Galerkin (DG) scheme, the currentcompact GKS uses the same local and compact stencil, achieves the 6th-order and8th-order accuracy, uses a much larger time step with CFL number $\geq 0.3$,and gets accurate solutions in both shock and smooth regions withoutintroducing trouble cell and additional limiting process. At the same time, thecurrent scheme solves the Navier-Stokes equations.

Compact Higher-order Gas-kinetic Schemes with Spectral-like Resolution for Compressible Flow Simulations