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

• Published in: Advances in computational mathematics 2024-04, Vol.50 (2), Article 21
• Main Authors: Ye, Xiao, Zheng, Xiangcheng, Liu, Jun, Liu, Yue
• Format: Article
• Language: English
• Subjects: Accuracy; Collocation methods; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Cost analysis; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Numerical analysis; Quadratic equations; Visualization
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-024-10116-9

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 L 2-1 σ formula on the graded mesh is used to account for the initial solution singularity, leading to an optimal QSC– L 2-1 σ 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.