Asymptotic-preserving approximations for stochastic incompressible viscous fluids and SPDEs on graph

Abstract The long-term dynamics of particles involved in an incompressible flow with a small viscosity ($\varepsilon>0$) and slow chemical reactions is depicted by a class of stochastic reaction-diffusion-advection (RDA) equations with a fast advection term of magnitude $1/\varepsilon $. It has been shown in Cerrai & Freidlin (2019) that the fast advection asymptotics of the 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 the AP property of the proposed approximation. Firstly, a strong error estimate, which depends on $1/\varepsilon $ linearly, is obtained via a variational argument. Secondly, we prove the consistency of exponential Euler approximations in the fast advection asymptotic regime between the original problem and the SPDE on the graph. Finally, a graph-weighted space is introduced to quantify the approximation error for the SPDE on the graph, which avoids the possible singularity near the vertices. Numerical experiments are carried out to support the theoretical results.