On the origin of numerical errors in the bounce-back boundary treatment of the lattice Boltzmann method: A remedy for artificial boundary slip and mass leakage

In this paper, we postulate a generalized bounce-back formulation, which can be simplified to the classic bounce-back method, the Ladd method (Ladd, 1994), and the midpoint method (Yin and Zhang, 2012) in the lattice Boltzmann method (LBM). The postulation process explicitly highlights the assumptions and constraints associated with these widely used boundary methods. Our demonstration simulations numerically confirm our analysis, and they also show that some artificial phenomena observed in previous studies, such as the boundary slip in Poiseuille flows and the mass leakage and apparent velocity in a hydrostatic system, are actually due to the violation of assumptions behind the formulations of these bounce-back methods. With proper modifications of these methods, these unphysical observations can be completely removed and the LBM solutions can match theoretical predictions up to the computer round-off resolution. Although the generalized bounce-back relation lacks practical usefulness, the analysis, simulations, and discussions in this work could be valuable for boundary method analysis, modification, development, and selection in LBM simulations of fluid flows and other processes.