Lattice Boltzmann method with diffusive scaling for space-fractional Navier-Stokes equations

Several conventional numerical methods, such as the finite difference method and finite element method, have been improved for solving space-fractional Navier-Stokes equations (SFNSEs). Nevertheless, the application of the lattice Boltzmann method (LBM) to solve the SFNSEs continues to pose a significant challenge. In this study, a fresh LBM is developed for solving the SFNSEs and the Maxwell iteration is carried out to validate the capacity of the present LBM in precisely recovering the macroscopic equations. The construction of the proposed LBM under diffusive scaling allows for the direct computation of fluid velocity from the distribution function without requiring any iterative procedures or approximations. Three numerical examples are implemented to assess the efficacy of the present LBM and validate its convergence order. The comparison between the numerical and analytical solutions reveals that the proposed LBM demonstrates second-order accuracy. The present LBM is employed to simulate the fractional Poiseuille flow, accompanied by a systematic exploration of the fractional flow characteristics within a channel. Compared with conventional numerical methods, the present LBM has advantages such as straightforward implementation, efficient parallelization, and easy handling of complex boundary conditions. The present LBM offers a new numerical approach to solve the SFNSEs, thereby facilitating the widespread application of the SFNSEs in science and engineering.