Finite-Difference Lattice Boltzmann Scheme for High-SpeedCompressible Flow: Two-Dimensional Case

Lattice Boltzmann (LB) modeling of high-speed compressible flows has long been attempted by various authors. One common weakness of most of previous models is the instability problem when the Mach number of the flow is large. In this paper we present a finite-difference LB model, which works for flows with flexible ratios of specificheats and a wide range of Mach number, from $0$ to 30 or higher. Besides the discrete-velocity-model by Watari [Physica A 382 (2007) 502], a modified Lax--Wendroff finite differencescheme and an artificial viscosity are introduced. The combination of the finite-difference scheme and the adding of artificial viscosity must find a balance of numerical stability versusaccuracy. The proposed model is validated by recovering results ofsome well-known benchmark tests: shock tubes and shock reflections.The new model may be used to track shock waves and/or to study thenon-equilibrium procedure in the transition between the regular andMach reflections of shock waves, etc.