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The stability and convergence analysis of finite difference methods for the fractional neutron diffusion equation

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Abstract

For the time-fractional neutron diffusion equation with a Caputo derivative of order \( \varvec{\alpha } \in ~(\varvec{0},\frac{\varvec{1}}{\varvec{2}}) \), we give the optimal error bounds of L1-type schemes under the spatial \( L^{\infty } \)-norm with lower regularity solution than typical \(| \varvec{\partial }_{\varvec{t}}^{\varvec{l}} \varvec{u}(\varvec{x},\varvec{t}) | \le \varvec{C} (\varvec{1+t}^{\varvec{2\alpha - l}}), \varvec{l = 0}, \varvec{1}, \varvec{2}\), where \(\varvec{2\alpha }\) is the highest order of the time-fractional derivative. The unique solvability and the numerical stability of schemes will be exported via a simple restriction of coefficients with non-uniform time steps \(\varvec{\tau }_{\varvec{n}} < \frac{\varvec{\pi }}{\varvec{4}}\). The truncation error of term \(\varvec{D}_{\varvec{0},\varvec{t}}^{\varvec{2\alpha }}\) with low regularity \(| \varvec{\partial }_{\varvec{t}}^{\varvec{l}} \varvec{u}(\varvec{x},\varvec{t}) | \le \varvec{C} (\varvec{1+t}^{\varvec{\alpha - l}}), \varvec{l = 0}, \varvec{1}, \varvec{2}\), and the global error of full discretizations will be given. In addition, several numerical examples will be shown to test our theoretical results.

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Data Availability

The data that support the finding of this study are available on request from the corresponding author upon reasonable request.

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Acknowledgements

The authors are grateful to the associate editor and two anonymous referees for their helpful comments.

Funding

The project is supported by the National Natural Science Foundation of China No.12171385.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, No. 28, Xianning West Road, Xi’an, 710049, Shaanxi, China

    Daopeng Yin, Yingying Xie & Liquan Mei

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  1. Daopeng Yin
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  2. Yingying Xie
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Correspondence to Liquan Mei.

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Communicated by: Martin Stynes

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Yin, D., Xie, Y. & Mei, L. The stability and convergence analysis of finite difference methods for the fractional neutron diffusion equation. Adv Comput Math 49, 72 (2023). https://doi.org/10.1007/s10444-023-10070-y

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