Approximation of the Levy-Feller advection-diffusion process by lattice Boltzmann method

In this paper, in order to expand the lattice Boltzmann method (LBM) to deal with more space-fractional systems, a fresh lattice Boltzmann scheme is proposed to approximate a Levy-Feller advection-diffusion process, which is governed by the Levy-Feller fractional advection-diffusion equation (LFADE). First, the fractional integral operator is discretized and the LFADE is transformed into a standard equation. Second, combining with Taylor expansion and Chapman-Enskog analysis, a family of the LFADE is recovered correctly from the continuous Boltzmann equation through selecting the equilibrium distribution functions. Finally, some test examples are presented and it is found that the numerical results agree well with the analytical solutions. In addition, the result in terms of stability is also tested by comparing with previous studies.