Steady State Convergence Acceleration of the Generalized Lattice Boltzmann Equation with Forcing Term through Preconditioning

TL;DR

This paper introduces a preconditioned generalized lattice Boltzmann method with forcing to significantly speed up steady state convergence in fluid flow simulations, especially at low Mach numbers, while improving numerical stability.

Contribution

The paper develops a preconditioned GLBE with forcing and multiple relaxation times, enabling faster convergence and enhanced stability in steady state flow computations with external forces.

Findings

Steady state convergence is accelerated by orders of magnitude.

Preconditioning improves numerical stability over single relaxation time methods.

Effective for flows with spatially and temporally varying external forces.

Abstract

Several applications exist in which lattice Boltzmann methods (LBM) are used to compute stationary states of fluid motions, particularly those driven or modulated by external forces. Standard LBM, being explicit time-marching in nature, requires a long time to attain steady state convergence, particularly at low Mach numbers due to the disparity in characteristic speeds of propagation of different quantities. In this paper, we present a preconditioned generalized lattice Boltzmann equation (GLBE) with forcing term to accelerate steady state convergence to flows driven by external forces. The use of multiple relaxation times in the GLBE allows enhancement of the numerical stability. Particular focus is given in preconditioning external forces, which can be spatially and temporally dependent. In particular, correct forms of moment-projections of source/forcing terms are derived such that they recover preconditioned Navier-Stokes equations with non-uniform external forces. As an illustration, we solve an extended system with a preconditioned lattice kinetic equation for magnetic induction field at low magnetic Prandtl numbers, which imposes Lorentz forces on the flow of conducting fluids. Computational studies, particularly in three-dimensions, for canonical problems show that the number of time steps needed to reach steady state is reduced by orders of magnitude with preconditioning. In addition, the preconditioning approach resulted in significantly improved stability characteristics when compared with the corresponding single relaxation time formulation.