Two time-stepping schemes for sub-diffusion equations with singular source terms
Abstract
Singular source terms in sub-diffusion equations may lead to the unboundedness of solutions, which will bring a severe reduction of convergence order of existing time-stepping schemes. In this work, we propose two efficient time-stepping schemes for solving sub-diffusion equations with a class of source terms mildly singular in time. One discretization is based on the Gr{\"u}nwald-Letnikov and backward Euler methods. First-order error estimate with respect to time is rigorously established for singular source terms and nonsmooth initial data. The other scheme derived from the second-order backward differentiation formula (BDF) is proved to possess second-order accuracy in time. Further, piecewise linear finite element and lumped mass finite element discretizations in space are applied and analyzed rigorously. Numerical investigations confirm our theoretical results.
Cite
@article{arxiv.2109.13473,
title = {Two time-stepping schemes for sub-diffusion equations with singular source terms},
author = {Han Zhou and Wenyi Tian},
journal= {arXiv preprint arXiv:2109.13473},
year = {2022}
}