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

TL;DR

This work develops an asymptotic-preserving exponential Euler scheme for a multiscale stochastic reaction-diffusion-advection equation with fast advection $1/\epsilon$, whose fast-limit is an SPDE on a graph induced by a Hamiltonian. By combining strong $L^{2}$-type error bounds that scale linearly with $1/\epsilon$, consistency between the SPDE in $\mathbb{R}^{2}$ and the SPDE on the graph, and a graph-based weighted $L^{2}$ framework to control vertex singularities, the authors prove that the AP scheme faithfully captures both the original multiscale dynamics and its graph limit. They show regularity and convergence properties for semigroups on the graph, and establish that the exponential Euler discretization preserves the fast-advection asymptotics, unlike standard Euler-Maruyama. Numerical experiments corroborate the predicted $\tfrac{1}{2}$ order in time and confirm convergence to the graph limit as $\epsilon\to0$, demonstrating the practicality of the AP method for SPDEs on graphs arising from multiscale flows. The results offer a robust route to simulate SPDEs on graphs and motivate future work on multi-critical Hamiltonians and fully discrete graph discretizations.

Abstract

The long-term dynamics of particles involved in an incompressible flow with a small viscosity ($ε>0$) and slow chemical reactions, is depicted by a class of stochastic reaction-diffusion-advection (RDA) equations with a fast advection term of magnitude $1/ε$. 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 $1/ε$ 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.

Figures (5)

• Figure 1: A commutative diagram.
• Figure 2: (a) Hamiltonian $H$ with three critical points corresponding to $O_1,O_2,O_3$; (b) Level sets of $H$; (c) Graph $\Gamma$ with three edges $I_1,I_2,I_3$ and four vertices $O_1,O_2,O_3,O_\infty$; (d) Level set $\{x\in\mathbb{R}^2,H(x)=H(O_3)\}$CF19.
• Figure 3: Errors of the exponential Euler approximation for \ref{['eq:SPDE-trun']} with $L=1$, $\epsilon=1$, $h=0.2$, $P=500$.
• Figure 4: Errors of the exponential Euler approximation for \ref{['eq:utzk-p-trun']} with $L=10$, $h=0.1$ and $\textup{P}=1000$.
• Figure 5: The asymptotic error $\mathbb E[\|U_{\epsilon}^{n}-(\bar{U}^{n})^{\vee}\|_{\mathbb{H}_\gamma}^2]$ with $\textup{P}=5000$, $\textup{Q}=100$, $\mathsf T=2^{-13}$, $\tau=2^{-18}$, $h=2$, $M_1=1$, $M_2=5$, $\psi\equiv 0$ and $(\widehat{u_1\mu})^\wedge(z)=10\sin(z)+6z$ against with $\epsilon=0.02:0.02:0.2.$

Theorems & Definitions (42)

• Example 2.1
• Example 2.2
• Lemma 2.3
• proof
• Example 2.4
• Example 2.5
• Lemma 3.1
• proof
• Remark 3.2
• Proposition 3.3