A third-order gas-kinetic scheme for three-dimensional inviscid and viscous flow computations

• A third-order multidimensional gas-kinetic scheme is given for three dimensional flows.
• High-order spatial and temporal accuracy are coupled nonlinearly in flux evaluation.
• Both normal and tangential derivatives of flow variables are included.
• Gaussian integration and Runge–Kutta method for space and time accuracy are avoided.
• The scheme is applied to both inviscid and viscous, and low and high speed flows.

With the WENO reconstruction, a third-order multidimensional gas-kinetic scheme, with inclusion of both normal and tangential derivatives of macroscopic flow variables, is constructed for the three-dimensional inviscid and viscous flow computations. For the flux evaluation, a time and space dependent gas distribution function from an initial piece-wise discontinuous polynomials around a cell interface is presented, where a multiple scale physical process from the kinetic to the hydrodynamic one is used for the flow evolution process. A high-order accuracy is achieved in the current scheme by integrating the space and time dependent flux function over the surface of a cell interface and a whole time step, without using the conventional Gaussian points integration for spatial accuracy and Runge–Kutta method for temporal accuracy, which have been used in many other high-order schemes. This scheme is applied to both inviscid and viscous, and low and high speed flow computations. The numerical tests clearly demonstrate that the current scheme is robust for the flows with strong discontinuities and accurate for viscous smooth flow solutions. The continuous effort on the gas kinetic scheme (GKS) is to develop a third method which has the same reliability and robustness as the well-developed second-order shock capturing schemes, but is simply much more accurate in all flow applications. The current scheme seems achieve such a target.