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

IF 1.7 3区数学 Q2 MATHEMATICS, APPLIED

Advances in Computational Mathematics Pub Date : 2024-03-22 DOI:10.1007/s10444-024-10116-9

Xiao Ye, Xiangcheng Zheng, Jun Liu, Yue Liu

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引用次数: 0

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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应用于非线性时间分数扩散方程的二维空间最佳二次样条配位法数值分析

最优二次样条拼合（QSC）方法因其高阶精度而被广泛应用于各种问题中，但由于扰动项会导致最优 QSC 离散的不对称等原因，相应的数值分析很少被研究。我们通过对一个非线性时间-分数扩散方程进行离散化，对二维空间的最优 QSC 方法进行数值分析，以作示范。梯度网格上的 L2-1 （_\sigma \）公式用于考虑初始解的奇异性，从而得出最优 QSC-L2-1 （_{\sigma }\ ）方案，其中非线性项由外推法处理。我们提供了数值解的存在性和唯一性，以及适当分级参数下的二阶时间精度和四阶空间精度。此外，我们还考虑了基于指数和技术的快速实现，以降低计算成本。我们进行了数值实验来验证理论分析和所提方案的有效性。

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来源期刊

Advances in Computational Mathematics 数学-应用数学

CiteScore

3.00

自引率

5.90%

发文量

审稿时长

3 months

期刊介绍： Advances in Computational Mathematics publishes high quality, accessible and original articles at the forefront of computational and applied mathematics, with a clear potential for impact across the sciences. The journal emphasizes three core areas: approximation theory and computational geometry; numerical analysis, modelling and simulation; imaging, signal processing and data analysis. This journal welcomes papers that are accessible to a broad audience in the mathematical sciences and that show either an advance in computational methodology or a novel scientific application area, or both. Methods papers should rely on rigorous analysis and/or convincing numerical studies.