Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise

Abstract

The stochastic time-fractional equation $\partial_t ψ-Δ\partial_t^{1-α} ψ= f + \dot W$ with space-time white noise $\dot W$ is discretized in time by a backward-Euler convolution quadrature for which the sharp-order error estimate \[ {\mathbb E}\|ψ(\cdot,t_n)-ψ_n\|_{L^2(\mathcal{O})}^2=O(τ^{1-αd/2}) \] is established for $α\in(0,2/d)$, where $d$ denotes the spatial dimension, $ψ_n$ the approximate solution at the $n^{\rm th}$ time step, and $\mathbb{E}$ the expectation operator. In particular, the result indicates optimal convergence rates of numerical solutions for both stochastic subdiffusion and diffusion-wave problems in one spatial dimension. Numerical examples are presented to illustrate the theoretical analysis.