The long-term dynamics of particles involved in an incompressible flow with a small viscosity ($\epsilon>0$) and slow chemical reactions, is depicted by a class of stochastic reaction-diffusion-advection (RDA) equations with a fast advection term of magnitude $1/\epsilon$. It has been shown in [7] the fast advection asymptotics of stochastic RDA equation in $\mathbb{R}^2$ can be characterized through a stochastic partial differential equation (SPDE) on the graph associated with certain Hamiltonian. To simulate such fast advection asymptotics, we introduce and study an asymptotic-preserving (AP) exponential Euler approximation for the multiscale stochastic RDA equation. There are three key ingredients in proving asymptotic-preserving property of the proposed approximation. First, a strong error estimate, which depends on $1/\epsilon$ linearly, is obtained via a variational argument. Second, we prove the consistency of exponential Euler approximations on the fast advection asymptotics between the original problem and the SPDE on graph. Last, a graph weighted space is introduced to quantify the approximation error for SPDE on graph, which avoids the possible singularity near the vertices. Numerical experiments are carried out to support the theoretical results.

翻译：对于具有小黏度（$\epsilon>0$）和慢化学反应的可压缩流中粒子的长期动力学，由一类带有量级为$1/\epsilon$的快平流项的随机反应-扩散-平流（RDA）方程刻画。文献[7]已证明，$\mathbb{R}^2$中随机RDA方程的快平流渐近性可通过与特定Hamiltonian相关的图上的随机偏微分方程（SPDE）来表征。为模拟此类快平流渐近性，我们提出并研究了一种针对多尺度随机RDA方程的渐近保持（AP）指数Euler逼近。证明该逼近渐近保持性质需三个关键要素：首先，通过变分论证获得线性依赖于$1/\epsilon$的强误差估计；其次，证明原问题与图上SPDE在快平流渐近性方面指数Euler逼近的一致性；最后，引入图加权空间量化图上SPDE的逼近误差，从而避免顶点附近的可能奇异性。数值实验验证了理论结果。