SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 2, Page 536-563, January 2021.
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.

中文翻译：

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随机对流扩散方程的格子玻尔兹曼方法

SIAM / ASA不确定性量化期刊，第9卷，第2期，第536-563页，2021年1月。
在本文中，我们为随机对流扩散方程（CDE）提出了一种格子Boltzmann方法（LBM）。采用随机Galerkin方法将随机CDE转换为确定性CDE系统，然后使用LBM离散确定性CDE。Maxwell迭代显示了该方法的一致性。由于确定性CDE的扩散系数矩阵是正定的，我们证明了LBM的加权$ L ^ 2 $-稳定性。通过这种稳定性，可以直接建立方法的收敛性。进行了数值实验，以验证LBM的准确性并证明其对随机CDE的有效性。