A Critical Study of the Compressible Lattice Boltzmann Methods for Riemann Problem

In this paper, the discrete velocity model proposed by Kataoka and Tsutahara (Phys. Rev. E 69(5):056702, 2004) for simulating inviscid flows is employed. Three approaches for improving the stability and the accuracy of this model, especially for high Mach numbers, are suggested and implemented in this research. First, the TVD scheme (Harten in J. Comput. Phys. 49:357---393, 1983) is used for space discretization of the convective term in the Lattice Boltzmann equation. Next, the modified Lax-Wendroff with artificial viscosity is employed to increase the robustness of the method in supersonic flows. Finally, a combination of TVD and the 2nd order derivative of the distribution function is employed using a differentiable switch. It is found that the recent technique is a more suitable approach for a wide range of Mach numbers. Moreover, the WENO scheme for space discretization has been applied and compared with these newly applied methods.