Residual-driven lattice Boltzmann methods: new transient and steady solvers

In this work, we propose a residual-driven lattice Boltzmann method (LBM).

Our objective is to enhance the LBM with advanced numerical algorithms

while maintaining its core simplicity—specifically, the streaming and collision

operations and low dissipation feature. This enhancement enables the

method to be applied efficiently, accurately, and easily to under-resolved

simulations, large-time-step computations, and steady-state problems. The

proposed approach demonstrates a speedup of 2 to 4 orders of magnitude for

2D lid-driven cavity flow, serving as a prototype for more complex scenarios.

Additionally, we provide theoretical predictions of computational efficiency,

although practical convergence rates reveal discrepancies. To address this

gap, we propose the nonlinear Krylov method and the boundary relaxation

technique. The LBM framework presents no fundamental barriers to being

extended by advanced numerical techniques; however, significant efforts are

still required to overcome conventional limitations.