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Volume 22, Issue 6
Modified BDF2 Schemes for Subdiffusion Models with a Singular Source Term

Minghua Chen, Jiankang Shi & Zhi Zhou

DOI: 10.4208/ijnam2025-1039

Int. J. Numer. Anal. Mod., 22 (2025), pp. 897-924.

Published online: 2025-08

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  • Abstract

The aim of this paper is to study the time stepping scheme for approximately solving the subdiffusion equation with a weakly singular source term. In this case, many popular time stepping schemes, including the correction of high-order BDF methods, may lose their high-order accuracy. To fill in this gap, in this paper, we develop a novel time stepping scheme, where the source term is regularized by using an $m$-fold integral-derivative and the equation is discretized by using a modified BDF2 convolution quadrature. We prove that the proposed time stepping scheme is second-order, even if the source term is nonsmooth in time and incompatible with the initial data. Numerical results are presented to support the theoretical results. The proposed approach is applicable for stochastic subdiffusion equation.

  • Keywords

Subdiffusion, modified BDF2 schemes, singular source term, error estimate.

  • AMS Subject Headings

26A33, 65M55, 65T50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-22-897, author = {Chen , MinghuaShi , Jiankang and Zhou , Zhi}, title = {Modified BDF2 Schemes for Subdiffusion Models with a Singular Source Term}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2025}, volume = {22}, number = {6}, pages = {897--924}, abstract = {

The aim of this paper is to study the time stepping scheme for approximately solving the subdiffusion equation with a weakly singular source term. In this case, many popular time stepping schemes, including the correction of high-order BDF methods, may lose their high-order accuracy. To fill in this gap, in this paper, we develop a novel time stepping scheme, where the source term is regularized by using an $m$-fold integral-derivative and the equation is discretized by using a modified BDF2 convolution quadrature. We prove that the proposed time stepping scheme is second-order, even if the source term is nonsmooth in time and incompatible with the initial data. Numerical results are presented to support the theoretical results. The proposed approach is applicable for stochastic subdiffusion equation.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2025-1039}, url = {http://global-sci.org/intro/article_detail/ijnam/24297.html} }
TY - JOUR T1 - Modified BDF2 Schemes for Subdiffusion Models with a Singular Source Term AU - Chen , Minghua AU - Shi , Jiankang AU - Zhou , Zhi JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 897 EP - 924 PY - 2025 DA - 2025/08 SN - 22 DO - http://doi.org/10.4208/ijnam2025-1039 UR - https://global-sci.org/intro/article_detail/ijnam/24297.html KW - Subdiffusion, modified BDF2 schemes, singular source term, error estimate. AB -

The aim of this paper is to study the time stepping scheme for approximately solving the subdiffusion equation with a weakly singular source term. In this case, many popular time stepping schemes, including the correction of high-order BDF methods, may lose their high-order accuracy. To fill in this gap, in this paper, we develop a novel time stepping scheme, where the source term is regularized by using an $m$-fold integral-derivative and the equation is discretized by using a modified BDF2 convolution quadrature. We prove that the proposed time stepping scheme is second-order, even if the source term is nonsmooth in time and incompatible with the initial data. Numerical results are presented to support the theoretical results. The proposed approach is applicable for stochastic subdiffusion equation.

Chen , MinghuaShi , Jiankang and Zhou , Zhi. (2025). Modified BDF2 Schemes for Subdiffusion Models with a Singular Source Term. International Journal of Numerical Analysis and Modeling. 22 (6). 897-924. doi:10.4208/ijnam2025-1039
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