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Numerical analysis for optimal quadratic spline collocation method in two space dimensions with application to nonlinear time-fractional diffusion equation

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Abstract

Optimal quadratic spline collocation (QSC) method has been widely used in various problems due to its high-order accuracy, while the corresponding numerical analysis is rarely investigated since, e.g., the perturbation terms result in the asymmetry of optimal QSC discretization. We present numerical analysis for the optimal QSC method in two space dimensions via discretizing a nonlinear time-fractional diffusion equation for demonstration. The L2-1\(_\sigma \) formula on the graded mesh is used to account for the initial solution singularity, leading to an optimal QSC–L2-1\(_{\sigma }\) scheme where the nonlinear term is treated by the extrapolation. We provide the existence and uniqueness of the numerical solution, as well as the second-order temporal accuracy and fourth-order spatial accuracy with proper grading parameters. Furthermore, we consider the fast implementation based on the sum-of-exponentials technique to reduce the computational cost. Numerical experiments are performed to verify the theoretical analysis and the effectiveness of the proposed scheme.

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Funding

The work of J. Liu was supported in part by the National Key R &D Program of China (No. 2023YFA1009003), the Shandong Provincial Natural Science Foundation (Nos. ZR2021MA020, ZR2020MA039), the Fundamental Research Funds for the Central Universities (Nos. 22CX03016A, 20CX05011A), and the Major Scientific and Technological Projects of CNPC under Grant (No. ZD2019-184-001). The work of X. Zheng was supported in part by the National Natural Science Foundation of China (No. 12301555), the National Key R &D Program of China (No. 2023YFA1008903), and the Taishan Scholars Program of Shandong Province (No. tsqn202306083).

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Authors and Affiliations

  1. College of Science, China University of Petroleum (East China), Qingdao, Shandong, 266580, China

    Xiao Ye, Jun Liu & Yue Liu

  2. School of Mathematics, Shandong University, Jinan, Shandong, 250100, China

    Xiangcheng Zheng

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  1. Xiao Ye
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  2. Xiangcheng Zheng
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  3. Jun Liu
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  4. Yue Liu
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Correspondence to Jun Liu.

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Communicated by: Martin Stynes

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Ye, X., Zheng, X., Liu, J. et al. Numerical analysis for optimal quadratic spline collocation method in two space dimensions with application to nonlinear time-fractional diffusion equation. Adv Comput Math 50, 21 (2024). https://doi.org/10.1007/s10444-024-10116-9

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