A one-step simplified lattice Boltzmann method without evolution of distribution functions

Recently, a simplified lattice Boltzmann method without evolution of distribution functions was developed. This method adopts the predictor-corrector scheme to evolve the macroscopic variables. Unlike the original simplified method, we develop a one-step scheme based on the resultant governing equations given by the Chapman-Enskog expansion analysis of the lattice Boltzmann equation. The proposed method abandons the predictor-corrector scheme and only needs one step to solve the governing equations without additional intermediate implementation. As a result, the computational efficiency can be improved. In addition, the theoretical analysis demonstrates that the present method has second-order of accuracy. To treat the boundary, the second-order approximations for equilibrium distribution functions are employed. To validate the proposed one-step method, several cases including the Poiseuille flow, the Womersley flow, the Taylor-Green vortex flow, and the lid-driven cavity flow are simulated. The numerical results agree with the analytical solutions or the reference data. And the second-order of accuracy is ensured. Moreover, simulations also suggest that our method has good performances on the numerical stability and the computational efficiency. And the improved efficiency is no less than 30% in our tests.