A stochastic Galerkin lattice Boltzmann method for incompressible fluid flows with uncertainties

Efficient modeling and simulation of uncertainties in computational fluid dynamics (CFD) remains
a crucial challenge. In this paper, we present the first stochastic Galerkin (SG) lattice Boltzmann
method (LBM) built upon the generalized polynomial chaos (gPC). The proposed method offers
an efficient and accurate approach that depicts the propagation of uncertainties in stochastic
incompressible flows. Formal analysis shows that the SG LBM preserves the correct Chapman–
Enskog asymptotics and recovers the corresponding macroscopic fluid equations. Numerical
experiments, including the Taylor–Green vortex flow, lid-driven cavity flow, and isentropic vortex
convection, are presented to validate the solution algorithm. The results demonstrate that the
SG LBM achieves the expected spectral convergence and the computational cost is significantly
reduced compared to the sampling-based non-intrusive approaches, e.g., the routinely used Monte
Carlo method. We obtain a speedup factor of 5.72 compared to Monte Carlo sampling in a
randomized two-dimensional Taylor–Green vortex flow test case. By leveraging the accuracy and
flexibility of LBM and the efficiency of gPC-based SG, the proposed SG LBM provides a powerful
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framework for uncertainty quantification in CFD practice.