Improving the low Mach number steady state convergence of the cascaded lattice Boltzmann method by preconditioning

Cascaded lattice Boltzmann method (LBM) involves the use of central moments in a multiple relaxation time formulation in prescribing the collision step. When the goal is to simulate low Mach number stationary flows, the greater the disparity between the magnitude of fluid speed and the sound speed, the higher is the numerical stiffness, which results in relatively large number of time steps for convergence of the LBM. One way to improve the steady state convergence of the scheme is to precondition the cascaded LBM, which reduces the disparities between the characteristic speeds or equivalently those between the eigenvalues of the system. In this paper we present a new preconditioned, two-dimensional cascaded LBM, where a preconditioning parameter is introduced into the equilibrium moments as well as the forcing terms. Particular focus is given to preconditioning differently the moments due to forcing at first and second orders so as to avoid any spurious effects in the emergent macroscopic equations. A Chapman–Enskog analysis performed on this approach shows consistency to the preconditioned Navier–Stokes equations. This modified central moment based scheme is then validated for accuracy by comparison against prior analytical or numerical results for certain benchmark problems, including those involving spatially variable body forces. Finally, significant steady state convergence acceleration of the preconditioned cascaded LBM is demonstrated for a set of characteristic parameters.