Lattice Boltzmann Method for Stochastic Convection-Diffusion Equations

In this paper, we propose a lattice Boltzmann method (LBM) for stochastic convection-diffusion equations (CDEs). The stochastic Galerkin method is employed to transform the stochastic CDE into a system of deterministic CDEs and the LBM is then used to discretize the deterministic CDEs. The consistency of the method is shown with the Maxwell iteration. Thanks to the property that the diffusion coefficient matrix of the deterministic CDEs is positive definite, we prove the weighted L-2-stability of the LBM. With this stability, the convergence of the method can be directly established. Numerical experiments are conducted to verify the accuracy of the LBM and demonstrate its effectiveness for stochastic CDEs. The numerical results not only are in good agreement with those existing in the literature but also show the ability of the LBM for stochastic problems with complex boundaries.