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A Kinetic Inviscid Flux (KIF) is proposed for simulating incompressible andcompressible flows. It is constructed based on the direct modeling ofmulti-scale flow behaviors, which is used in the Gas-Kinetic Scheme (GKS), theUnified Gas-Kinetic Scheme (UGKS), the Discrete Unified Gas-Kinetic Scheme(DUGKS), etc.. In KIF, the discontinuities (such as the shock wave) that cannot be well resolved by mesh cells are mainly solved by the Kinetic Flux VectorSplitting (KFVS) method representing the free transport mechanism (ormicro-scale mechanism), while in other flow regions that are smooth, the flowbehavior is solved mainly by the central-scheme-like Totally ThermalizedTransport (TTT). The weights of KFVS and TTT in KIF is automatically determinedby those in the theory of direct modeling. Two ways of choosing the weights inKIF are proposed, which are actually the weights adopted in the UGKS and theDUGKS, respectively. By using the test cases of Sod shock tube, rarefactionwave, the boundary layer of flat plate, the cavity flow and hypersonic flowover circular cylinder, the validity and accuracy of the present method areexamined. The KIF does not suffer from the carbuncle phenomenon, and does notintroduce extra numerical viscosity in smooth regions. Especially, in the caseof hypersonic cylinder, it gives a quite sharp and clear density andtemperature contours. The KIF can be viewed as an inviscid-viscous splittingversion of the GKS. By the doing this splitting, it is easy to be used in thetraditional CFD frameworks. It can also be classified as a new type in thenumerical schemes based on the kinetic theory that are represented by the worksin Ref. \cite{SunA} and Ref. \cite{ohwada2018a}, except the weights aredetermined by the weights of direct modeling.

A multi-scale kinetic inviscid flux extracted from the gas-kinetic scheme for simulating incompressible and compressible flows