Lattice Boltzmann Method for Variable-order Time-fractional Navier-Stokes Equation

Lattice Boltzmann (LB) method for solving the variable-order time-fractional Navier-Stokes equation is developed. Firstly, by discretizing the historical part and the current part of the variable-order fractional derivative respectively, the variable-order time-fractional Navier-Stokes equation is transformed into an integer-order partial differential equation. To improve the computational efficiency, in addition to using the L1 direct discretization method to discretize the historical part of the fractional derivative term, a fast estimation method is also adopted for calculation. Subsequently, LB model based on the two-dimensional nine-velocity (D2Q9) lattice model is constructed, and the Chapman-Enskog analysis is carried out to determine the specific expression of the equilibrium distribution function. It is theoretically proved that this LB model can accurately derive the target macroscopic equation. Finally, through numerical examples, it is verified that the constructed LB model can accurately solve the variable-order time-fractional Navier-Stokes equation, and the spatial convergence order of this LB model is approximately of the second order.