Sharp Error Analysis for Averaging Crank-Nicolson Schemes with Corrections for Subdiffusion with Nonsmooth Solutions

Abstract

Thanks to the singularity of the solution of linear subdiffusion problems, most time-stepping methods on uniform meshes can result in

O(\tau )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(\tau )$$\end{document}

accuracy where

\tau

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau $$\end{document}

denotes the time step. The present work aims to discover the reason why some type of Crank-Nicolson schemes (the averaging Crank-Nicolson (ACN) scheme) for the subdiffusion can only yield

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(\tau ^\alpha )\,\, (\alpha <1)$$\end{document}

accuracy, which is much lower than the desired. The existing well-developed error analysis for the subdiffusion, which has been successfully applied to many time-stepping methods such as the fractional BDF-

p(1⩽p⩽6)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\,\,(1\leqslant p\leqslant 6)$$\end{document}

, requires singular points to be out of the path of contour integrals involved. The ACN scheme in this work is quite natural but fails to meet this requirement. By resorting to the residue theorem, some novel sharp error analysis is developed in this study, upon which correction methods are further designed to obtain the optimal

O({\tau }^2)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(\tau ^2)$$\end{document}

accuracy. All results are verified by numerical tests.

Keywords

Subdiffusion / Uniform mesh / Crank-Nicolson scheme / Convolution quadrature / Singularity

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Funding

National Natural Science Foundation of China(12201322)

Natural Science Foundation of Inner Mongolia Autonomous Region(2020MS01003)

Autonomous Region Level High-Level Talent Introduction Research Support Program in 2022(12000-15042224)

RIGHTS & PERMISSIONS

Shanghai University