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 \[ {\mathbb E}\|ψ(\cdot,t_n)-ψ_n\|_{L^2(\mathcal{O})}^2=O(τ^{1-αd/2}) \] is established for $\alpha \in (0,2/d)$, where $d$ denotes the spatial dimension, ${\psi }_n$ the approximate solution at the $n^{\mathrm{th}}$ 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.

References in corpus (3)

• Analysis of $L1$-Galerkin FEMs for time-fractional nonlinear parabolic problems
• Time-stepping error bounds for fractional diffusion problems with non-smooth initial data
• Convergence of finite element solutions of stochastic partial integro-differential equations driven by white noise