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


Authors: Max Gunzburger, Buyang Li and Jilu Wang
Journal: Math. Comp. 88 (2019), 1715-1741
MSC (2010): Primary 60H15, 60H35, 65M12
DOI: https://doi.org/10.1090/mcom/3397
Published electronically: November 27, 2018
MathSciNet review: 3925482
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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

$\displaystyle ({\mathbb{E}}\Vert\psi (\cdot ,t_n)-\psi _n\Vert _{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.

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Additional Information

Max Gunzburger
Affiliation: Department of Scientific Computing, Florida State University, Tallahassee, Florida 32306
Email: gunzburg@fsu.edu

Buyang Li
Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
Email: buyang.li@polyu.edu.hk

Jilu Wang
Affiliation: Department of Scientific Computing, Florida State University, Tallahassee, Florida 32306
Address at time of publication: Department of Mathematics and Statistics, Mississippi State University, Starkville, Mississippi 39762
Email: jwang@math.mssate.edu

DOI: https://doi.org/10.1090/mcom/3397
Keywords: Stochastic partial differential equation, time-fractional derivative, space-time white noise, error estimates
Received by editor(s): January 31, 2017
Received by editor(s) in revised form: January 5, 2018, April 10, 2018, and August 2, 2018
Published electronically: November 27, 2018
Additional Notes: The research of the first and third authors was supported in part by the USA National Science Foundation grant DMS-1315259 and by the USA Air Force Office of Scientific Research grant FA9550-15-1-0001.
The work of the second author was supported in part by the Hong Kong RGC grant 15300817.
Article copyright: © Copyright 2018 American Mathematical Society