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

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.