哈密顿快速平流含多个临界点下指数欧拉方法的渐近保持性
Asymptotic preservation of an exponential Euler method under Hamiltonian fast advection with multiple critical points
查看机构详情
- The Hong Kong Polytechnic University（香港理工大学）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
本文证明指数欧拉方法在含多个临界点的快速哈密顿平流下具有渐近保持性，通过加权L²框架和解析半群估计获得接近1/2的时间收敛阶，并数值验证。
中文摘要 AI 辅助
我们建立了在 $\mathbb R^2$ 中、具有多个临界点的快速哈密顿平流下，随机反应-扩散-平流方程的指数欧拉方法的渐近保持性质。在快速平流极限下，动力学简化为非紧度量图上的随机偏微分方程。一个关键要素是极限指数欧拉格式的强收敛性分析，该分析因图的非紧性、其多条边和顶点、以及图算子的非一致椭圆性和顶点退化而变得复杂。利用加权 $L^2$ 框架和解析半群平滑估计，我们证明了对于加权 $L^2$ 初始数据，时间收敛阶任意接近 $1/2$，并且在半阶分数域条件下精确为 $1/2$。将这些估计与快速平流极限相结合，在多个临界点情形下得到了渐近保持性质。数值实验展示了渐近行为并确认了预测的时间收敛速率。
英文摘要
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.
