Numerical stability analysis for semi-implicit FDLBM

We analyze the numerical stability of Finite Difference Lattice Boltzmann Method (FDLBM) by means of von Neumann stability analysis. A FDLBM is proposed to ensure the numerical stability by relaxing the Lagragian particle convection and by satisfying CFL condition. The stability boundary of FDLBM depends on the BGK relaxation time, the CFL number, the mean flow velocity, and the wavenumber. We show that stability regions, as a function of the relaxation time and of the CFL number, drastically change as varying the difference method, which discretizes the kinetic equation. With the centerered difference, FDLBM has characteristics opposite to the conventional LBM, that is, as the BGK relaxation time is increased, stability region of the FDLBM is decreased. We also analyze the numerical stability of FDLBM capable of simulating multi-phase flow by additional fictitious forcing terms. The use of semi-implicit schemes improves numerical stability. With semi-implicit upwind difference, though CFL number equals to unity, we can simulate phase transition without encountering any numerical stability problem.