Asymptotic preservation of an exponential Euler method under Hamiltonian fast advection with multiple critical points

Abstract

We establish the asymptotic-preserving property of the exponential Euler method for stochastic reaction-diffusion-advection equations in $\mathbb R^2$ under fast Hamiltonian advection with multiple critical points. In the fast-advection limit, the dynamics reduces to a stochastic partial differential equation on a noncompact metric graph. A key ingredient is the strong convergence analysis of the limiting exponential Euler scheme, which is complicated by the graph's noncompactness, its multiple edges and vertices, and the nonuniform ellipticity and vertex degeneracy of the graph operator. Using a weighted $L^2$ framework and analytic-semigroup smoothing estimates, we prove a temporal convergence order arbitrarily close to $1/2$ for weighted $L^2$ initial data and exactly $1/2$ under a half-order fractional-domain condition. Combining these estimates with the fast-advection limits yields the asymptotic-preserving property in the multiple-critical-point setting. Numerical experiments illustrate the asymptotic behavior and confirm the predicted temporal convergence rate.