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

Abstract:

The stochastic time-fractional equation $\partial _t \psi -\Delta \partial _t^{1-\alpha } \psi = f + \dot W$ with space-time white noise $\dot W$ is discretized in time by a backward-Euler convolution quadrature for which the sharp-order error estimate \[ ({\mathbb E}\|\psi (\cdot ,t_n)-\psi _n\|_{L^2(\mathcal {O})}^2)^{\frac {1}{2}}=O(\tau ^{\frac {1}{2}-\frac {\alpha d}{4}}) \] is established for $\alpha \in (0,2/d)$, where $d$ denotes the spatial dimension, $\psi _n$ the approximate solution at the $n$th time step, and $\mathbb {E}$ the expectation operator. In particular, the result indicates sharp 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.