  The goal of this paper is to develop a high order numerical method based on
Kinetic Inviscid Flux (KIF) method and Flux Reconstruction (FR) framework. The
KIF aims to find a balance between the excellent merits of Gas-Kinetic Scheme
(GKS) and the lower computational costs. The idea of KIF can be viewed as an
inviscid-viscous splitting version of the gas-kinetic scheme, and Shu and
Ohwada have made the fundamental contribution. The combination of Totally
Thermalized Transport (TTT) scheme and Kinetic Flux Vector Splitting (KFVS)
method are achieved in KIF. Using a coefficient which is related to time step
$\delta t$ and averaged collision time $\tau$, KIF can adjust the weights of
TTT and KFVS flux in the simulation adaptively. By doing the inviscid-viscous
splitting, KIF is very suitable and easy to integrate into the existing
framework. The well understood FR framework is used widely for the advantages
of robustness, economical costs and compactness. The combination of KIF and FR
is originated by three motivations. The first purpose is to develop a high
order method based on the gas kinetic theory. The second reason is to keep the
advantages of GKS. The last aim is that the designed method should be more
efficient. In present work, we use the KIF method to replace the Riemann flux
solver applied in the interfaces of elements. The common solution at the
interface is computed according to the gas kinetic theory, which makes the
combination of KIF and FR scheme more reasonable and available. The accuracy
and performance of present method are validated by several numerical cases. The
Taylor-Green vortex problem has been used to verify its potential to simulate
turbulent flows.
