A High-order Gas-Kinetic Navier-Stokes Solver I: One-dimensional Flux Evaluation

The foundation for the development of modern compressible flow solver is the Riemann solution of the inviscid Euler equations. The high-order schemes are basically related to high-order spatial interpolation or reconstruction. Due to the first-order wave interaction in the Riemann solution, the temporal accuracy is improved through the Runge-Kutta method, where the dynamic deficiencies in the 1st-order Riemann solution is alleviated through the sub-step spatial reconstruction in the Runge-Kutta process. The close coupling between the spatial and temporal evolution in the original nonlinear governing equations seems weakened due to its spatial and temporal decoupling. A solution under piecewise discontinuous high-order initial reconstruction for the Navier-Stokes equations directly is on a urgent need for many high-order methods. Unfortunately, the piece-wise discontinuous initial data and the hyperbolic-parabolic nature of the Navier-Stokes equations seem to be inconsistent mathematically, such as the divergence of the viscous and heat flux due to discontinuity. In this paper, based on the Boltzmann equation, we are going to present a flux function for the Navier-Stokes equations starting from a high-order reconstruction. The theoretical validity for such an approach is due to the fact that the Boltzmann equation has no specific requirement on the smoothness of the initial data and the kinetic equation has the mechanism to describe the gas evolution starting from an initial discontinuous data. The current high-order flux evaluation method is an extension of the second-order gas-kinetic BGK scheme for the Navier-Stokes equations (BGK-NS). The novelty for the easy extension from a 2nd-order, which is equivalent to the Generalized Navier-Stokes flow solver, to an even higher order is solely due to the simple particle transport mechanism on the microscopic level, i.e., the particle free transport and collisions. This paper will present a hierarchy to construct such a high-order method. Numerical examples for a 3rd-order scheme will be presented. The necessity to couple spatial and temporal evolution nonlinearly in the flux evaluation can be clearly observed through the numerical performance of the scheme for the viscous flow computations. email:[email protected] email:[email protected]