Lattice BGK model for time-fractional incompressible Navier–Stokes equations

Abstract

In this paper, a novel D2Q9 lattice Boltzmann model with BGK operator (LBGK) is proposed for the incompressible time-fractional Navier–Stokes equations with Caputo-type fractional derivative. First the fractional derivative is divided into the history part and the local part, in which the former is approximated using the efficient algorithm for the evaluation of the Caputo fractional derivative, while the latter is simply approximated by rectangle formula to keep the time-dependent characteristics of Navier–Stokes equations as the evolution equations. Then, a LBGK model is constructed for Navier–Stokes equations after approximation of the Caputo derivative. Through Chapman–Enskog analysis the macroscopic equations can be recovered from this model in the small Mach number limit. At the end of this paper, a numerical example with analytic solutions is carried out to show that the LBGK model is efficient.