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arXiv · 2109.13473

Two time-stepping schemes for sub-diffusion equations with singular source terms

Han Zhou·Wenyi Tian
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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.

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Han Zhou, Wenyi Tian. 2021-09-28. Two time-stepping schemes for sub-diffusion equations with singular source terms. https://doi.org/10.1007/s10915-022-01914-8

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