Evaluation of gas-kinetic schemes for solving 1d inviscid compressible flow problems

In this paper, two classes of gas-kinetic schemes are investigated for one-dimensional compressible inviscid flow, namely, the Kinetic Flux Vector Splitting (KFVS) scheme and the Bhatnagar-Gross-Krook (BGK) scheme. Second-order high-resolution scheme for both methods are also developed for calculating flows containing discontinuities. This is achieved by means of reconstructing the initial data via Monotone Upstream-Centered Schemes for Conservation Laws (MUSCL) approach. The Total Variation Diminishing (TVD) shock capturing properties of these second-order schemes are achieved through the use of non-linear limiters. In addition, a multistage TVD Runge-Kutta method is employed for the time integration of the finite volume gas-kinetic scheme. Exclusive consideration is focused on the BGK scheme, which yields a better numerical result in comparison with the KFVS scheme. Three typical one-dimensional inviscid flow problems containing shocks, namely, steady flow in a divergent nozzle, the unsteady shock tube problem, and two interacting blast waves problem are analyzed numerically with the gas-kinetic schemes. Numerical results from the first-order and second-order gas-kinetic schemes are presented and compared with the exact solutions. Other computed results such as those from the Steger-Warming's Flux Vector Splitting scheme, Roe's Flux Difference Splitting scheme, MacCormack's scheme, and high-order compact Van Leer's Flux Splitting Scheme are also presented as comparisons to the gas-kinetic schemes. In addition, the effects of grid sizes on the numerical results of the gas-kinetic schemes are also investigated.