A stable least squares finite difference-based fractional step lattice Boltzmann method for incompressible flows

In this work, a fractional step lattice Boltzmann method (FS-LBM) enhanced by the least-squares finite difference (LSFD) approach is proposed to simulate incompressible flows. The conventional FS-LBM consists of prediction and correction steps, with the latter typically involving the solution of an anti-diffusion equation that requires discretizing the Laplacian operator. Previous studies have employed discretization schemes such as the central difference (CD) stencil and the finite difference stable stencil (SS). While the SS stencil improves numerical stability, it does so at the cost of considerably reduced accuracy compared with the CD stencil. To achieve a better balance between accuracy and stability, the LSFD method is introduced to discretize the Laplacian operator. The proposed method is validated through two- and three-dimensional isothermal and thermal flows. Numerical results demonstrate that the FS-LBM with LSFD maintains second-order convergence, offers superior stability compared to both CD and SS stencils, and achieves higher accuracy than the SS stencil. Furthermore, analysis of weighting functions within the LSFD framework shows that an appropriate choice of weighting function can significantly enhance accuracy, with the W 2 weighting function performing best in 2D cases and W 5 in 3D cases.