With the development of high-speed computers and the demand from challenging problems in engineering and industries, high-order numerical methods are prevailing in computational fluid dynamics. Most current high-order schemes for the Euler and Navier-Stokes equations are based on the Riemann solver to evaluate the flux and the multi-stage Runge-Kutta time stepping techniques for temporal discretization. Different from them, the gas-kinetic scheme (GKS) is based on the Bhatnagar-Gross-Krook model which gives a time-dependent evolution solution. The high-order GKS (HGKS) has been systematically developed in the past decades, with good performance in continuum flow simulations. The current HGKS are efficient, accurate, and robust. Although the HGKS within the finite volume framework has ac...[ Read more ]

With the development of high-speed computers and the demand from challenging problems in engineering and industries, high-order numerical methods are prevailing in computational fluid dynamics. Most current high-order schemes for the Euler and Navier-Stokes equations are based on the Riemann solver to evaluate the flux and the multi-stage Runge-Kutta time stepping techniques for temporal discretization. Different from them, the gas-kinetic scheme (GKS) is based on the Bhatnagar-Gross-Krook model which gives a time-dependent evolution solution. The high-order GKS (HGKS) has been systematically developed in the past decades, with good performance in continuum flow simulations. The current HGKS are efficient, accurate, and robust. Although the HGKS within the finite volume framework has achieved great success, there are few schemes based on the difference formulation. In this thesis, we will aim to construct the HGKS under difference frameworks for the Euler and Navier-Stokes solutions.

Firstly, the conservative finite difference HGKS is constructed for inviscid and viscous flows. Rather than the use of the discontinuous gas distribution function in the finite volume framework, the continuous fluxes at the node points will be kinetically split and then utilized to achieve high-order spatial accuracy through the reconstruction. The fifth-order WENO-Z reconstruction will be implemented on those kinetic-splitting fluxes to improve the resolution of the scheme. The two-stage fourth-order (S2O4) time stepping method, together with the second-order GKS flux, is used to have high-order temporal discretization. Numerical tests for flows with vortexes and shocks have validated the robustness and accuracy of the new method. For the finite volume one using the same WENO-Z reconstruction, S2O4 time-stepping method, and the GKS flux, the CPU time is 4 times as much as that of the finite difference scheme. And the efficiency tests regarding the CPU time vs. the errors show that the current scheme is more efficient than the same order finite volume version of HGKS as well. However, the above finite difference GKS cannot be directly extended to the unstructured mesh because the conservative property cannot be satisifed automatically. Thus, we turn to a new class of methods called spectral difference methods.

Then spectral difference gas-kinetic scheme (SDGKS) has been successfully developed on the unstructured quadrilateral meshes. It's high-order, conservative, and efficient, with the advantages of both finite volume and finite difference methods, such as flexibility in geometry and efficiency in computation. The S2O4 time marching technique is also adopted since the derivatives required in S2O4 can be easily obtained by the time-accurate GKS flux function. In this way, compared with the traditional RK methods, 60% of iteration stages are saved. The efficiency is higher than the traditional schemes for viscous flows. Stability analysis for the linear advection equation shows that the new scheme is linearly stable. The numerical simulations agree well with the analytic and reference solutions.

The new difference gas-kinetic schemes in this thesis, are generally high-order accurate, efficient, robust, and conservative, with possible further improvement in the future.