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Second-order error analysis of the averaged L1 scheme L1 for time-fractional initial-value and subdiffusion problems

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Science China Mathematics 2024年第7期67卷 1641-1664页

作者： Jinye Shen Fanhai Zeng martin.stynes School of Mathematics Southwestern University of Finance and EconomicsChengdu611130China School of Mathematics Shandong UniversityJinan250100China Applied and Computational Mathematics Division Beijing Computational Science Research CenterBeijing100193China

Fractional initial-value problems(IVPs) and time-fractional initial-boundary value problems(IBVPs), each with a Caputo temporal derivative of order α ∈(0, 1), are considered. An averaged variant of the well-known L1 scheme is proved to be O(N^(-2)) convergent for IVPs on suitably graded meshes with N points, thereby improving the O(N^(-(2-α))) convergence rate of the standard L1 scheme. The analysis relies on a delicate decomposition of the temporal truncation error that yields a sharp dependence of the order of convergence on the degree of mesh grading used. This averaged L1 scheme can be combined with a finite difference or piecewise linear finite element discretization in space for IBVPs, and under a restriction on the temporal mesh width, one gets again O(N^(-2)) convergence in time, together with O(h^(2)) convergence in space,where h is the spatial mesh width. Numerical experiments support our results.

关键词： time-fractional subdiffusion averaged L1 scheme

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A Survey of the L1 Scheme in the Discretisation of Time-Fractional Problems

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Numerical Mathematics(Theory,Methods and Applications) 2022年第4期15卷 1173-1192页

作者： martin.stynes Applied and Computational Mathematics Division Beijing Computational Science Research CenterBeijing 100193China

A survey is given of convergence results that have been proved when the L1 scheme is used to approximate the Caputo time derivative D_(t)^(α)(where 0<α<1)in initial-boundary value problems governed by D_(t)^(α)u−△u=f and similar equations,while taking into account the weak singularity that is present in typical solutions of such *** aspects of these analyses are outlined,such as global and local convergence bounds and the techniques used to derive them,fast implementation of the L1 scheme,semilinear problems,multi-term time derivatives,α-robustness,a posteriori error analysis,and two modified L1 schemes that achieve better *** fifty references are provided in the bibliography,more than half of which are from the period 2019-2022.

关键词： L1 scheme fractional derivative survey

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A Parameter-Uniform Finite Difference Method for a Coupled System of Convection-Diffusion Two-Point Boundary Value Problems

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Numerical Mathematics(Theory,Methods and Applications) 2008年第2期1卷 176-197页

作者： Eugene O'Riordan Jeanne stynes martin.stynes School of Mathematical Sciences Dublin City University Glasnevin Department of Computing Cork Institute of Technology Department of Mathematics National University of Ireland

A system of m (≥2) linear convection-diffusion two-point boundary value problems is examined,where the diffusion term in each equation is multiplied by a small parameterεand the equations are coupled through their convective and reactive terms via matrices B and A *** system is in general singularly perturbed. Unlike the case of a single equation,it does not satisfy a conventional maximum princi- *** hypotheses are placed on the coupling matrices B and A that ensure exis- tence and uniqueness of a solution to the system and also permit boundary layers in the components of this solution at only one endpoint of the domain;these hypotheses can be regarded as a strong form of diagonal dominance of *** solution is decomposed into a sum of regular and layer *** are established on these compo- nents and their derivatives to show explicitly their dependence on the small parameterε.Finally,numerical methods consisting of upwinding on piecewise-uniform Shishkin meshes are proved to yield numerical solutions that are essentially first-order conver- gent,uniformly inε,to the true solution in the discrete maximum *** results on Shishkin meshes are presented to support these theoretical bounds.

关键词：两点边值问题对流扩散异常扰动连接系统

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