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

被引:0
作者
Xiao Ye
Xiangcheng Zheng
Jun Liu
Yue Liu
机构
[1] China University of Petroleum (East China),College of Science
[2] Shandong University,School of Mathematics
来源
Advances in Computational Mathematics | 2024年 / 50卷
关键词
Nonlinear fractional diffusion equation; Optimal quadratic spline collocation method; Nonuniform ; 2-1; formula; Convergence analysis; Fast implementation; 65M12; 65M15; 65M70;
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摘要
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 L2-1σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_\sigma $$\end{document} formula on the graded mesh is used to account for the initial solution singularity, leading to an optimal QSC–L2-1σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\sigma }$$\end{document} 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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  • [1] Alikhanov AA(2015)A new difference scheme for the time fractional diffusion equation J. Comput. Phys. 280 424-438
  • [2] Bialecki B(2008)Optimal superconvergent one step quadratic spline collocation methods BIT 48 449-472
  • [3] Fairweather G(1994)Quadratic spline collocation methods for elliptic partial differential equations BIT 34 33-61
  • [4] Karageorghis A(2010)Quadratic spline collocation for one-dimensional parabolic partial differential equations Numer. Algorithm 53 511-553
  • [5] Nguyen QN(2006)Optimal quadratic and cubic spline collocation on nonuniform partitions Computing 76 227-257
  • [6] Christara CC(2019)Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem J. Sci. Comput. 79 624-647
  • [7] Christara CC(2020)Why fractional derivatives with nonsingular kernels should not be used, Fract. Calc Appl. Anal. 23 610-634
  • [8] Chen T(2002)Analysis of fractional differential equations J. Math. Anal. Appl. 265 229-248
  • [9] Dang DM(2002)CTRW pathways to the fractional diffusion equation Chem. Phys. 284 13-27
  • [10] Christara CC(1988)Quadratic-spline collocation methods for two-point boundary value problems Internat. J. Numer. Methods Engrg. 26 935-952