High-order BDF convolution quadrature for stochastic fractional evolution equations driven by integrated additive 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 $O(\tau^{\alpha + \gamma -1/2})$ for $1< \alpha + \gamma \leq 5/2$, and $O(\tau^{2})$ for $5/2< \alpha + \gamma <3$, where $\alpha \in (1, 2)$ and $\gamma \in (0, 1)$ denote the time fractional order and the order of the integrated noise, respectively. Furthermore, this convergence rate could be improved to $O(\tau^{\alpha + \gamma -1/2})$ for any $\alpha \in (1, 2)$ and $\gamma \in (0, 1)$, if we employ the ID$3$-BDF3 method. The argument could be easily extended to the subdiffusion model with $\alpha \in (0, 1)$. Numerical examples are provided to support and complement the theoretical findings.