High-order BDF convolution quadrature for stochastic fractional evolution equations driven by integrated additive noise
Minghua Chen, Jiankang Shi, Zhen Song, Yubin Yan, Zhi Zhou
TL;DR
The paper tackles numerical approximation of linear stochastic time-fractional evolution equations driven by integrated additive noise, where the nonlocal operator and noise limit regularity and degrade standard time-stepping accuracy. It introduces a smoothing approach via an ID$m$-fold integral-differential regularization of the noise, combined with high-order BDF convolution quadrature to form the ID$m$-BDF$k$ schemes. The authors establish rigorous mean-square error bounds showing that ID$2$-BDF$2$ achieves $O(\tau^{α+γ-1/2})$ convergence for $1< α+γ\le 5/2$ (and $O(\tau^{2})$ for $5/2< α+γ<3$), while ID$3$-BDF$3$ yields $O(τ^{α+γ-1/2})$ for all $α∈(1,2), γ∈(0,1)$, with extensions to subdiffusion $α∈(0,1)$. Numerical experiments corroborate the theory and demonstrate practical effectiveness for stochastic subdiffusion and diffusion-wave models. The smoothing strategy and high-order convolution-quadrature framework offer a significant improvement over existing low-order schemes for stochastic time-fractional problems with integrated noise.
Abstract
The numerical analysis of stochastic time fractional evolution equations presents considerable challenges due to the limited regularity of the model caused by the nonlocal operator and the presence of noise. The existing time-stepping methods exhibit a significantly low order convergence rate. In this work, we introduce a smoothing technique and develop the novel high-order schemes for solving the linear stochastic fractional evolution equations driven by integrated additive noise. Our approach involves regularizing the additive noise through an $m$-fold integral-differential calculus, and discretizing the equation using the $k$-step BDF convolution quadrature. This novel method, which we refer to as the ID$m$-BDF$k$ method, is able to achieve higher-order convergence in solving the stochastic models. Our theoretical analysis reveals that the convergence rate of the ID$2$-BDF2 method is $O(τ^{α+ γ-1/2})$ for $1< α+ γ\leq 5/2$, and $O(τ^{2})$ for $5/2< α+ γ<3$, where $α\in (1, 2)$ and $γ\in (0, 1)$ denote the time fractional order and the order of the integrated noise, respectively. Furthermore, this convergence rate could be improved to $O(τ^{α+ γ-1/2})$ for any $α\in (1, 2)$ and $γ\in (0, 1)$, if we employ the ID$3$-BDF3 method. The argument could be easily extended to the subdiffusion model with $α\in (0, 1)$. Numerical examples are provided to support and complement the theoretical findings.